8: Further Applications of Trigonometry - Mathematics

In this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of problems, including finding the height of a tree. We extend topics we introduced in Trigonometric Functions and investigate applications more deeply and meaningfully.

• 8.0: Prelude to Further Applications of Trigonometry
The world’s largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California. Just how do scientists know its true height? A common way to measure the height involves determining the angle of elevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This method is much more practical than climbing the tree and dropping a very long tape measure.
• 8.1: Non-right Triangles - Law of Sines
In this section, we will find out how to solve problems involving non-right triangles. The Law of Sines can be used to solve oblique triangles. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side. There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution.
• 8.2: Non-right Triangles - Law of Cosines
Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angle is between two known sides, a SAS (side-angle-side) triangle, or when all three sides are known, but no angles are known, a SSS (side-side-side) triangle. In this section, we will investigate another tool for solving oblique triangles described by these last two cases.
• 8.3: Polar Coordinates
When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled (r,θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
• 8.4: Polar Coordinates - Graphs
polar equation describes a relationship between rr and θ on a polar grid. It is easier to graph polar equations if we can test the equations for symmetry. There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry.
• 8.5: Polar Form of Complex Numbers
In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.
• 8.6: Parametric Equations
We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.
• 8.7: Parametric Equations - Graphs
In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.
• 8.8: Vectors
Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s ground speed and bearing, while investigating another approach to problems of this type.
• 8.E: Further Applications of Trigonometry (Exercises)
• 8.R: Further Applications of Trigonometry (Review)

Introduction to The Unit Circle: Sine and Cosine Functions

Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. Similarly, the progression from day to night occurs as a result of Earth’s rotation, and the pattern of the seasons repeats in response to Earth’s revolution around the sun. Outside of nature, many stocks that mirror a company’s profits are influenced by changes in the economic business cycle.

In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions.

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Contents

The term "trigonometry" derives from the Greek "τριγονομετρία" ("trigonometria"), meaning "triangle measuring", from "τρίγονο" (triangle) + "μετρειν" (to measure).

Our modern word "sine", is derived from the Latin word sinus, which means "bay" or "fold", from a mistranslation (via Arabic) of the Sanskrit word jiva, alternatively called jya. Ώ] Aryabhata used the term ardha-jiva ("half-chord"), which was shortened to jiva and then transliterated by the Arabs as jiba ( جب ). European translators like Robert of Chester and Gherardo of Cremona in 12th-century Toledo confused jiba for jaib ( جب ), meaning "bay", probably because jiba ( جب ) and jaib ( جب ) are written the same in the Arabic script (this writing system uses accents instead of vowels and in some formats the accents are not written to ease writing, so if the readers are not familiar with the language they might be confused between words with the same letters but different phonetics). The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae. ΐ] These roughly translate to "first small parts" and "second small parts".

Tables of natural functions

To be of practical use, the values of the trigonometric functions must be readily available for any given angle. Various trigonometric identities show that the values of the functions for all angles can readily be found from the values for angles from 0° to 45°. For this reason, it is sufficient to list in a table the values of sine, cosine, and tangent for all angles from 0° to 45° that are integral multiples of some convenient unit (commonly 1′). Before computers rendered them obsolete in the late 20th century, such trigonometry tables were helpful to astronomers, surveyors, and engineers.

For angles that are not integral multiples of the unit, the values of the functions may be interpolated. Because the values of the functions are in general irrational numbers, they are entered in the table as decimals, rounded off at some convenient place. For most purposes, four or five decimal places are sufficient, and tables of this accuracy are common. Simple geometrical facts alone, however, suffice to determine the values of the trigonometric functions for the angles 0°, 30°, 45°, 60°, and 90°. These values are listed in a table for the sine, cosine, and tangent functions.

1. Graphs, Functions, and Models

1.1 Introduction to Graphing

1.3 Linear Functions, Slope, and Applications

1.4 Equations of Lines and Modeling

1.5 Linear Equations, Functions, Zeros, and Applications

1.6 Solving Linear Inequalities

2.1 Increasing, Decreasing, and Piecewise Functions Applications

2.2 The Algebra of Functions

2.3 The Composition of Functions

2.6 Variation and Application

3. Quadratic Functions and Equations Inequalities

3.2 Quadratic Equations, Functions, Zeros, and Models

3.3 Analyzing Graphs of Quadratic Functions

3.4 Solving Rational Equations and Radical Equations

3.5 Solving Equations and Inequalities with Absolute Value

4. Polynomial Functions and Rational Functions

4.1 Polynomial Functions and Modeling

4.2 Graphing Polynomial Functions

4.3 Polynomial Division The Remainder Theorem and the Factor Theorem

4.4 Theorems about Zeros of Polynomial Functions

4.6 Polynomial Inequalities and Rational Inequalities

5. Exponential Functions and Logarithmic Functions

5.2 Exponential Functions and Graphs

5.3 Logarithmic Functions and Graphs

5.4 Properties of Logarithmic Functions

5.5 Solving Exponential Equations and Logarithmic Equations

5.6 Applications and Models: Growth and Decay Compound Interest

6. The Trigonometric Functions

6.1 Trigonometric Functions of Acute Angles

6.2 Applications of Right Triangles

6.3 Trigonometric Functions of Any Angle

6.4 Radians, Arc Length, and Angular Speed

6.5 Circular Functions: Graphs and Properties

6.6 Graphs of Transformed Sine Functions and Cosine Functions

7. Trigonometric Identities, Inverse Functions, and Equations

7.1 Identities: Pythagorean and Sum and Difference

7.2 Identities: Cofunction, Double-Angle, and Half-Angle

7.3 Proving Trigonometric Identities

7.4 Inverses of the Trigonometric Functions

7.5 Solving Trigonometric Equations

8. Applications of Trigonometry

8.3 Complex Numbers: Trigonometric Notation

8.4 Polar Coordinates and Graphs

8.5 Vectors and Applications

9. Systems of Equations and Matrices

9.1 Systems of Equations in Two Variables

9.2 Systems of Equations in Three Variables

9.3 Matrices and Systems of Equations

9.6 Determinants and Cramer&rsquos Rule

9.7 Systems of Inequalities and Linear Programming

10. Analytic Geometry Topics

10.2 The Circle and the Ellipse

10.4 Nonlinear Systems of Equations and Inequalities

10.6 Polar Equations of Conics

11. Sequences, Series, and Combinatorics

11.2 Arithmetic Sequences and Series

11.3 Geometric Sequences and Series

11.4 Mathematical Induction

11.5 Combinatorics: Permutations

11.6 Combinatorics: Combinations

8: Further Applications of Trigonometry - Mathematics

The Improving Mathematics Education in Schools (TIMES) Project

Measurement and Geometry : Module 23Year : 9-10

• Familiarity with Pythagoras’ theorem.
• Basic knowledge of congruence and similarity of triangles.
• Knowledge of the basic properties of triangles, squares and rectangles.
• Facility with simple algebra and equations.
• Familiarity with the use of a calculator.

The word trigonometry signifies the measurement of triangles and is concerned with the study of the relationships between the sides and angles in a triangle. We initially restrict our attention to right-angled triangles.

Trigonometry was originally developed to solve problems related to astronomy, but soon found applications to navigation and a wide range of other areas. It is of great practical importance to builders, architects, surveyors and engineers and has many other applications.

Suppose we lean a ladder against a vertical wall. By moving the ladder closer to the wall, thereby increasing the angle between the ladder and the ground, we increase the distance up the wall that the ladder can reach. Since the length of the ladder remains the same, Pythagoras’ theorem relates the distance up the wall to the distance of the ladder from the base of the wall. Trigonometry allows us to relate that same distance to the angle between the ladder and the ground.

To measure the height of a flagpole, an observer can measure a distance out from the base and the angle made between the top of the pole and observer’s eye as shown in
the diagram.

As with many topics in Mathematics, trigonometry is a subject which continues on into the senior years of secondary school and well beyond into higher mathematics. Modern telecommunications depend on an understanding and harnessing of signal processing, which is modelled by the trigonometric functions.

Each right-angled triangle contains a right angle. The congruence tests tell us that
if either of the following pieces of information is known, then the triangle is
completely determined:

• one other angle and one side (by the ASA congruence test)
• two sides (by either the SAS test or the RHS test).

Trigonometry and Pythagoras’ theorem enable us to find the remaining sides and angles in both cases.

If we have two similar right-angled triangles (sometimes called right triangles ), then the angles of one match up with the angles of the other and their matching sides are in the same ratio.

For example, the following triangles are similar.

Since the matching sides are in the same ratio,

 = = .

This means that the ratio

 = = = .
 = = = and = = = .

That is, once the angles of a triangle are fixed, the ratios of the sides of the triangle are constant. In a right-angled triangle, we only need to know one other angle and then the angle sum of a triangle gives us the third angle. Hence, in a right-angled triangle, if we know one other angle, then the ratios of the sides of the triangle are constant.

This is the basis of trigonometry.

Draw up an angle of 58° using a protractor. Place markers at distances of 3cm, 5cm and 8cm and draw perpendiculars as shown in the diagram.

Measure the heights marked, h 1 , h 2 , h 3 and calculate to one decimal place the ratios , , .

Why are the ratios (approximately) equal ?

In order to distinguish the various possible ratios in a right-angled triangle, we introduce some names. We always refer to the longest side (opposite the right-angle) as the hypotenuse .

We now choose one of the two acute angles and label it, often using one of the Greek letters &alpha , &beta , or &theta . We shall call this angle the reference angle .

The side opposite the reference angle is called the opposite side , generally referred to as the opposite, and the remaining side is called the adjacent side , or simply adjacent, since it is next to the reference angle.

As mentioned above, once we fix the size of the reference angle in a right-angled triangle then the ratios of various sides remain the same irrespective of the size of the triangle. There are six possible trigonometric ratios we could use. We mainly work with just three of them. The remaining three are their reciprocals.

The ratio of the opposite to the adjacent is known as the tangent ratio or the tangent of the angle &theta . (The name comes from an earlier time and involves the use of circles.) We write

 tan &theta = where 0° < &theta < 90°.

Write down the value of tan &theta .

 a b

 a tan &theta = b tan &theta = =

By drawing an appropriate triangle explain why tan 45° = 1

From the first exercise above, you will have found that the ratios , , were all approximately 1.6. These ratios correspond to a tangent ratio, so tan 58° is approximately 1.6. The calculator gives tan 58° &asymp 1.6000334529.

The tangent ratio for other angles can be found using a calculator. Needless to say the calculator does not ‘draw a triangle’ but uses clever mathematical algorithms to find the value. It is important that students make sure their calculator is in degree mode .

We can use the tangent ratio to calculate missing sides in a right-angled triangle, provided we are given the right information.

 a Find the length marked x in the diagram below, correct to two decimal places. b Find x , correct to two decimal places

 a tan 33° = b tan 43° = Hence x = 13 tan 33° Hence x tan 43° = 11 &asymp 8.44 so x = &asymp 11.80

The Three BASIC Trigonometric Ratios

In addition to the tangent ratio, there are two other basic ratios that we use.
These are known as the sine and cosine ratios. We take &theta to be the reference angle
so 0° < &theta < 90°. The three ratios are defined by:

sin &theta = , cos &theta = , tan &theta =

Students need to learn these definitions thoroughly. One simple mnemonic that might assist them is SOH CAH TOA, consisting of the first letter of each ratio and the first letter of the sides making up that ratio.

For the following triangle, write down the value of:

 a sin &theta b cos &theta c tan &theta

 a sin &theta = b cos &theta = c tan &theta = = = = = = =

The angles 30°, 45° and 60° appear frequently in trigonometry and their sine, cosine and tangent ratios can be expressed using rational numbers and surds.

In the diagrams below, calculate all sides and angles and then fill in the table below.

 &theta 30° 45° 60° sin &theta cos &theta tan &theta

We can use the trigonometric ratios to calculate missing sides in a right-angled triangle.

Find the value of x , correct to four decimal places.

 a b c

a Explain why sin (90° &minus &theta ) = cos &theta , and simplify cos (90° &minus &theta ).

b Prove that = tan &theta .

c It is standard notation to write (sin &theta ) 2 as sin 2 &theta , and (cos &theta ) 2 as cos 2 &theta .

Prove that cos 2 &theta + sin 2 &theta = 1.

d Explain why 0 < cos &theta < 1 and 0 < sin &theta < 1.

We have seen that in any right-angled triangle with reference angle &theta , there are three basic ratios associated with that angle and that the value of each ratio can be found using a calculator. In order to use the trigonometric ratios to find angles in a right-angled triangle, we need to reverse the process. Thus, given the sine, cosine or tangent of some angle between 0 and 90 degrees, we want to find the angle with the given ratio.

We have seen that tan 45° = 1. We say that the inverse tangent of 1 is 45°. This is written as tan -1 1 = 45°.

Students must not confuse this &minus1 index with its usual algebraic meaning of ‘one over’.
To help avoid this, it is best to read the symbol tan -1 as inverse tangent and not as tan to the minus one .

Similarly, since sin 30° = 0.5, we write sin -1 0.5 = 30° and say: the inverse sine of 0.5 is 30° .

To find, for example, cos -1 0.25, we use the calculator, which gives 75.52°, correct to two decimal places.

Find each of the following, correct to the nearest degree.

 a sin -1 0.6 b cos -1 0.412 c tan -1 2 d the angle &theta for which sin &theta = 0.8 e the angle &theta for which cos &theta = 0.2

 a sin -1 0.6 = 36.86989. ° b cos -1 0.412 = 65.66946. ° &asymp 37° &asymp 66° c tan -1 2 = 63.43494. ° d If sin &theta = 0.8 &asymp 63° then &theta = sin -1 0.8 = 53.13. ° &asymp 53° e If cos &theta = 0.2 then &theta = cos -1 0.2 = 78.463. ° &asymp 78°

The following exercise shows us how to find an angle from two given sides.

a Find the angle between the diagonal and the base of the rectangle, correct to the nearest degree.

b Find the base angle in the isosceles triangle,
correct to the nearest degree.

Angles of Elevation and Depression

When an observer looks at an object that is higher than the (eye of) the observer, the angle between the line of sight and the horizontal is called the angle of elevation .

On the other hand, when the object is lower than the observer, the angle between the horizontal and the line of sight is called the angle of depression . These angles are always measured from the horizontal.

From the top of a cliff, 100 m above sea level, the angle of depression to a ship sailing past is 17°. How far is the ship from the base of the cliff, to the nearest metre?

The diagram shows the top of the cliff P , the ship S and the base of the cliff B .
Let SB = x m be the distance of the ship from the cliff. By alternate angles, PSB = 17°.

 Hence tan 17° = so x = = 327.085

The distance is 327 m (to the nearest metre).

In some problems several steps may be required to find the desired lengths.

From a point A , 10 m from the base of a tree, the angle of elevation of the top of a tree is 36°. From a point B , x m further on from the base of the tree, the angle of elevation is 20°.

a Find the height of the tree to the nearest tenth of a metre.

b Find the distance x m to the nearest tenth of a metre.

 a Let the height of the tree be h metres In OHA , tan 36° = so h = 10 tan 36° = 7.2654. &asymp 7.3 m, to the nearest tenth of a metre

Note: The approximation 7.3 is not used in part b. It leads to the wrong answer.

 b In HOB , = tan 20° hence OB = = = 19.96. so x = AB = OB &minus OA = 9.96. &asymp 10.0 m to the nearest tenth of a metre

Consider the diagram shown opposite.

a Prove that BD = ( AD + CD ).

b Hence deduce that

tan DEB = tan AED + tan CED

One of the early applications of trigonometry was to improve navigation. Bearings are used to indicate the direction of an object (perhaps a ship) from a fixed reference point (perhaps a light house). True bearings give the angle &theta ° from the north measured clockwise. We write the true bearing of &theta ° as &theta ° T, where &theta ° is an angle between 0° and 360°. It is customary to write the angle using three digits, so 0° T is written as 000° T, and a true bearing of 15° is written as 015° T.

We can use trigonometry to solve problems involving bearings.

Anthony walks for 490 metres on a bearing of 140° T from point A to point B .

Find to the nearest metre,

a how far east the point B is from the point A .

b how far south the point B is from the point A .

 a Distance east = 490 sin 40° &asymp 315 m (correct to the nearest metre) b Distance south = 490 cos 40° &asymp 375 m (correct to the nearest metre)

Navigational questions can also be given using compass bearings.

From a point P , a ship sails N 20° E for 20km to the point Q. From Q the ship sails N 70° W until it is at the point R directly north of P . Find the distance PR .

If we take any triangle with two given sides a and b about a given (acute) angle , then the area of the triangle is given by

Area = ab sin &theta

By dropping an altitude in the triangle shown below, derive the formula stated above.

Note that the above formula works even in the case when &theta is obtuse. In the module, Further Trigonometry, we will show how to define the sine of an obtuse angle.

From the material developed so far it should be apparent that the trigonometric ratios are a powerful tool for relating angles and sides in right-angled triangles. Not all triangles, of course, possess a right angle. We can extend the angle/side connection to an arbitrary triangle using two basic formulas known as the sine rule and the cosine rule .

The sine rule states that in the triangle shown,

= = .

In words this says that in any triangle, any side divided by the sine of the opposite angle is equal to any other side divided by the sine of its opposite angle.

The cosine rule states that in the triangle shown

c 2 = a 2 + b 2 &minus 2 ab cos C .

This may be thought of as Pythagoras’ theorem with a correction term.

You may care to derive these from the material established in this module. The sine rule is relatively easy to prove. For the cosine rule, drop a perpendicular from A and apply Pythagoras’ theorem to the two triangles. The details of the proofs are given in the module, Further Trigonometry and in the module, Pythagoras’ Theorem.

In general, a triangle may contain an obtuse angle as well as acute angles. Thus far we have only defined the trigonometric ratios for acute angles.

The trigonometric ratios can be extended to include obtuse angles and this will be done in the module, Further Trigonometry .

If &theta is an obtuse angle then the sine of &theta is given by

sin &theta = sin (180° &minus &theta ).

Note that if &theta is an obtuse angle, then its supplement, 180° &minus &theta , is acute.

Similarly, if &theta is an obtuse angle then the cosine of &theta is given by

cos &theta = &minus cos (180° &minus &theta ),

and the tangent of &theta is given by

tan &theta = &minus tan (180° &minus &theta ).

Thus, the sine of an obtuse angle equals the sine of its supplement and the cosine and tangent of an obtuse angle equal the negative of the cosine and tangent of its supplement, respectively.

In the module, Trigonometric Functions we extend the definitions even further to include all angles, including those greater than 180° and negative angles.

Once we are able to find the sine of all possible angles, we can plot the graph of the function
y = sin x . This leads us to study the trigonometric functions which are used to describe wave motion and electronic signal processing.

There are six possible ratios that one might take between the three sides of a right-angled triangle. We can take the reciprocal of the three ratios sine, cosine and tangent to produce the remaining three.

These ratios are called: cosecant, secant and cotangent respectively. These are written as
cosec &theta , sec &theta , and cot &theta .

 cosec &theta = = , sec = &theta = , cot &theta = =

The prefix co- is short for complementary, since the cosine of an angle is equal to the sine of its complement. The cotangent of an angle is the tangent of its complement and similarly the cosecant of an angle is the secant of its complement.

We saw in the examples in this module, that we can get by with using only the sine, cosine and tangent ratios in problem solving. However, the reciprocal ratios arise naturally when we study the calculus of the trigonometric functions.

 a 1 + tan 2 &theta = sec 2 &theta . b cot 2 &theta + 1 = cosec 2 &theta .

The Plimpton 322 tablet, believed to have been written about 1800BC, contains a table of numbers in base 60 that are believed by some to be an early record of tangent ratios. This remains controversial.

The Greeks developed a form of trigonometry based on chords in a circle. Hipparchus of Nicaea (c. 180-125 BC), sometimes called `the father of Trigonometry’, came from near present day Istanbul. He was an astronomer and he was able to calculate the duration of the year to within 6 minutes. To assist with his astronomical calculations, he developed an early form of trigonometry and drew up tables of chords. This table was later extended, and used, by Ptolemy (c.90&minus160 AD).

Hipparchus divided the circumference of a circle into 360° (as the Babylonians had done) and the diameter into 120 parts.

Then for a given arc AB subtending an angle />at the centre, he gives the length of the corresponding chord crd ( />), as shown in the diagrams below.

Note then that crd (60°) = R and also crd (90°) = R .

The length of the chord is related to the sine ratio as the following diagram shows.

AB = crd( ) = 2 R sin .

Hipparchus did his calculations with R = 60. Algebraic notation was not available at this time.

With the decline of civilisation in the West, Indian and Arab mathematicians built on the work of the Greeks. In about the 5th century AD, in a work known as Surya Siddhanta, a table of sines appeared. The Arabs also developed spherical trigonometry, again to be used in astronomy.

As we saw above, the sine is related to half a chord, and the word sine itself, arose from a series of mistranslations of a Sanskrit word for half-chord. The Indian word was transliterated into Arabic and then misunderstood as a word meaning breast . This in turn came to mean curve and hence bay or gulf which is sinus in Latin.

The motivation for much of early trigonometry was astronomy. The first pure trigonometrical text written in Europe was De triangulis Omnimodis (‘On all sorts of triangles’) by Johann Müller (1436 &minus 1476), alias Regiomontanus. His work was motivated by his translation of Ptolemy, in which he realised that a more systematic treatment of trigonometry was required. He wrote:

You who wish to study such great and wonderful things, who wonder at the

In his work, he derives and proves the sine rule.

The cosine was defined by cos &theta = sin (90° &minus &theta ). This is the sine of the complement of the angle.

The terms tangent and secant were introduced by T. Fink (1561 &minus 1656) while cotangent, (the complement of the tangent), was introduced by by E. Gunther (1581 &minus1626).

The following diagram shows how the names tangent and secant arose:

AB = tan , OB = sec .

If the radius of the circle is 1, then AB , which has length tan &alpha , touches the circle at A
(Latin tango &minus I touch) and OB , which has length sec &alpha , cuts the circle (Latin seco &minus I cut.)

The right-angled triangles are similar to each other.

Draw a right-angled isosceles triangle. The equal angles are each 45°.

 &theta 30° 45° 60° sin &theta cos &theta tan &theta 1

a , b and c are the side lengths of triangle ABC . />C = 90° and />A = &theta °.

a sin &theta = and cos B = . But B = 90° &minus &theta ° . Therefore cos (90 &minus &theta ) = sin &theta .

b tan &theta = = ÷ = .

c sin 2 &theta + cos 2 &theta = + = = 1

d The hypotenuse is the longest side and so sin &theta = < 1 and cos &theta = < 1.

a 30° b 44°

 a BD = AD &minus AB BD = BC + CD Adding gives 2 BD = AD + CD BD = ( AD + CD ) b tan DEB = = = tan AED + tan CED

Draw BD perpendicular to AC with D on AC .

 Area = b × BD = ab sin &theta

a Divide both sides by cos 2 &theta to obtain the result.

b Divide both sides by sin 2 &theta to obtain the result

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations.

The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.

Hands-on Activity Trig River

Units serve as guides to a particular content or subject area. Nested under units are lessons (in purple) and hands-on activities (in blue).

Note that not all lessons and activities will exist under a unit, and instead may exist as "standalone" curriculum.

• Where Is Here?
• Nidy-Gridy: Using Grids and Coordinates
• Northward Ho! Create and Use Simple Compasses
• How to be a Great Navigator!
• Vector Voyage!
• The North (Wall) Star
• Navigating by the Numbers
• Stay in Shape
• Trig River
• Accuracy, Precision and Errors in Navigation: Getting It Right!
• Close Enough? Angles & Accuracy of Measurement in Navigation
• Computer Accuracy
• Sextant Solutions
• Topo Map Mania!
• The Trouble with Topos
• Getting to the Point
• Classroom Triangles
• Topo Triangulation
• Triangulate: Topos, Compasses and Triangles, Oh My!
• You've Got Triangles!
• Navigational Techniques by Land, Sea, Air and Space
• Navigating at the Speed of Satellites
• GPS on the Move
• Making GPS Art: Draw It, Walk It, Log It, Display It!
• GPS Scavenger Hunt
• Not So Lost in Space
• A Roundabout Way to Mars
• Satellite Tracker

How wide is a river?

Engineering Connection

Sometimes engineers cannot directly measure an object's size or distance because it would take too much time, or it is physically impossible (a tape measure to find the distance from the Earth to Pluto?). Instead of actually measuring a size or distance, engineers use trigonometry and other mathematical relationships to estimate it very accurately.

Learning Objectives

After this activity, students should be able to:

• Use right triangle trigonometry and angle measurements to calculate distances
• Convert from US customary to metric units
• Perform averaging and comparison of numbers
• Explain how engineers use trigonometry and other mathematical relationships to estimate distances

Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source e.g., by state within source by type e.g., science or mathematics within type by subtype, then by grade, etc.

Common Core State Standards - Math
• Use units as a way to understand problems and to guide the solution of multi-step problems choose and interpret units consistently in formulas choose and interpret the scale and the origin in graphs and data displays. (Grades 9 - 12) More Details

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International Technology and Engineering Educators Association - Technology
• Knowledge gained from other fields of study has a direct effect on the development of technological products and systems. (Grades 6 - 8) More Details

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State Standards
• Solve real-world and mathematical problems involving the four operations with rational numbers. (Grade 7) More Details

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Introduction/Motivation

Is it possible to determine the width of a river without crossing it? (Answer: It is possible to come very close to determining the width of a river of any size, using triangles.) The same principle used to determine the width of a river can be applied to other situations, including determining the height of a hill, a tree or a building. The simple geometric shape that makes this all possible is the triangle. During this activity, you will learn how to use triangles to determine the width of a river.

Procedure

Trigonometry is a branch of mathematics dealing with relationships between the angles and sides of triangles. The three basic trigonometric relations that we are concerned with in this activity are: sine, cosine, and tangent (abbreviated as sin, cos, and tan). They are the ratios of the lengths of two sides of a particular triangle. The type of triangle that is most useful to mathematics is a right triangle, which has one angle equal to 90º. By definition, the 90º-angle is made by two lines that are perpendicular to each other (like the corner of a square), and a sloping line connecting the two perpendicular lines makes the third side of the triangle. This sloping line is called the hypotenuse, and the name comes from the Greek hypo (meaning under) and teinein (meaning to stretch). It is easiest to show this visually:

The letters SOH CAH TOA can help students remember which sides go with which functions (Sine = Opposite / Hypotenuse, etc.). Mnemonics may help 6-8th graders memorize the relations: "Some Old Hag Caught A Hippie Tripping On Art" or "Some Oaf Happily Cut A Hole Through Our Apartment."

• Print out enough Trig River Worksheets for each student and the Paper Half Protractors, if enough protrators are not available for students to use/share.
• Determine whether to conduct the activity indoors or outside.
• Prepare or choose objects that can be used as markers and shore boundaries.

Is it possible to determine the width of a river without crossing it? (Allow discussion and entertain any creative ideas: walk through the river, throw a rope across, use stepping-stones, etc. Then state that this river that does not allow solutions those solutions because it is too deep, the current is too swift, it is too wide, you do not have that tool, etc.) If a student knows about using triangles, have them explain as much as they can or introduce the idea and give a quick review of a right triangle — drawing and labeling it on the board.

Define a "river" for the students. For example, if working inside, rearrange desks to form the two "banks" of the river (with space for students to work on each "shore"). If working outside, choose a spot with two widely spaced (2-5 meters) and roughly parallel lines to define the "river" banks. For example, a wide sidewalk, two lines on a football field, or a strip of grass with straight edges. If a small "river" is being measured, have the students measure in centimeters, if a larger "river" is being used, have the students measure in meters. Because these are ratios of distance, the result at the end should have the same unit (meters, centimeters, etc.) used to make the initial measurement.

1. On one side of the river (as close to the middle of that side as possible, set an object that will be the Far Edge Marker. Normally this represents a tree right at the edge of the opposite side of the river.
2. Directly across from the marker, place a Zero Edge Marker (see Figure 1). All the students should be on this side of the river.
3. Lay the measuring tape along the "zero edge bank" with one end at the Zero Edge Marker and place a piece of tape every ½ meter on the river edge of the desks. Repeat this in the other direction (see Figure 1).
4. Give each student a worksheet.
5. Each student should make an estimate of how wide the river is and record it on the back of their worksheet (anywhere on the paper is acceptable).
6. Each group will work from a different tape mark. When both students of a group are at their disignated mark and have written on their worksheet the distance their tape is from the Zero Edge Marker, give each group a protractor.

Figure 1. Set up for angle measurement.

1. Lay the protractor with the center point on the middle of the tape and the zero angle pointing toward the Zero Edge Marker (see Figure 2).
2. One student will hold the protractor in place while the other places one end of the string on the center point of the protractor and aims the other end at the Far Edge Marker. Read the angle the string passes over on the protractor (counting up from zero this should not be more than 90 degrees), and record it on the worksheet. While the students do this, the teacher can measure the actual distance between the two markers do not reveal the distance yet.
3. Partners switch jobs and record a second measurement on their worksheet.
4. Complete the worksheet calculations. (Leave the desks and markers in place.)
5. Have students compare their estimate of the river width to the actual measurement. How close was their estimate?
6. Have the students use both metric and English units to measure the distance from the zero marker. Compare the two results at the end.

Question and Answer #1: Were students who were closer to the zero marker more or less accurate than those further away? (Answer: Students close to the zero marker should be less accurate because the values of the tangents of angles close to 90º become large quickly and a small error in the angle measurement results in a large distance error. Note that the same problem would be seen as the measured angle approached zero degrees, but a student would have to be infinitely far away for that.)

Question and Answer #2: Could this measuring method be used in the wilderness if you did not have a calculator or Trig Tables? (Answer: It is not easy to memorize the tangent values for all angles but one value is very easy to remember: tan(45). Have the students find this value and then explain why they get a simple answer.)

Figure 2. Angle Measurement

Assessment

Discussion Question: Solicit, integrate and summarize student responses.

• Is it possible to determine the width of a river without crossing it? (Allow discussion and entertain any creative ideas: walk through the river, throw a rope across, use stepping-stones, etc. Then state that this river does not allow those solutions because it is too deep, the current is too swift, it is too wide, you do not have that tool, etc.) See "With the Students" in the Procedure section.

Prediction: Have the students estimate how wide the river is and record predictions on the board.

Activity Embedded Assessment

Worksheet: Have the students complete the activity worksheet, Trig Calculations review their answers to gauge their mastery of the subject.

• Were students closer to the zero marker more or less accurate than those further away? (Answer: Students close to the zero marker should be less accurate because the values of the tangents of angles close to 90º become large quickly and a small error in the angle measurement results in a large distance error. Note that the same problem would be seen as the measured angle approached zero degrees, but a student would have to be infinitely far away for that.)
• Could this method be used in the wilderness if you did not have a calculator or Trig Tables? (Answer: It is not easy to memorize the tangent values for all angles but one value is very easy to remember: tan(45). Have the students find this value and then explain why they get a simple answer.)

Safety Issues

• Have students use caution not to bump heads when grouping up on taped marks they may have to take turns if their tape marks are too close for a group of students.
• To avoid tripping over desk legs, remind students to move carefully between the "banks" of the imaginary river.

Troubleshooting Tips

Students may get confused on the trigonometry involved in this activity. After all of the students have tried to figure out the distance across the river, choose one of the tape marks and have the class walk through the activity together for reinforcement.

Right Triangle Trigonometry

• Use right triangles to evaluate trigonometric functions.
• Find function values forand
• Use equal cofunctions of complementary angles.
• Use the deﬁnitions of trigonometric functions of any angle.
• Use right-triangle trigonometry to solve applied problems.

Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.

Using Right Triangles to Evaluate Trigonometric Functions

(Figure) shows a right triangle with a vertical side of lengthand a horizontal side has lengthNotice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle.

Figure 1.

We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle, x. (Adjacent means “next to.”) The opposite side is the side across from the angle, y. The hypotenuse is the side of the triangle opposite the right angle, 1. These sides are labeled in (Figure).

Figure 2. The sides of a right triangle in relation to angle

Given a right triangle with an acute angle ofthe first three trigonometric functions are listed.

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”

For the triangle shown in (Figure), we have the following.

How To

Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.

1. Find the sine as the ratio of the opposite side to the hypotenuse.
2. Find the cosine as the ratio of the adjacent side to the hypotenuse.
3. Find the tangent as the ratio of the opposite side to the adjacent side.

Evaluating a Trigonometric Function of a Right Triangle

Given the triangle shown in (Figure), find the value of

Figure 3.

The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17.

Try It

Given the triangle shown in (Figure), find the value of

Reciprocal Functions

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.

Take another look at these definitions. These functions are the reciprocals of the first three functions.

When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in (Figure). The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

Figure 5. The side adjacent to one angle is opposite the other angle.

Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.

How To

Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.

1. If needed, draw the right triangle and label the angle provided.
2. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
3. Find the required function:
• sine as the ratio of the opposite side to the hypotenuse
• cosine as the ratio of the adjacent side to the hypotenuse
• tangent as the ratio of the opposite side to the adjacent side
• secant as the ratio of the hypotenuse to the adjacent side
• cosecant as the ratio of the hypotenuse to the opposite side
• cotangent as the ratio of the adjacent side to the opposite side

Evaluating Trigonometric Functions of Angles Not in Standard Position

Using the triangle shown in (Figure), evaluate

Figure 6.

Analysis

Another approach would have been to find sine, cosine, and tangent first. Then find their reciprocals to determine the other functions.

Try It

Using the triangle shown in (Figure),evaluate

Figure 7.

Finding Trigonometric Functions of Special Angles Using Side Lengths

It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples ofandRemember, however, that when dealing with right triangles, we are limited to angles between

Suppose we have atriangle, which can also be described as atriangle. The sides have lengths in the relationThe sides of atriangle, which can also be described as atriangle, have lengths in the relationThese relations are shown in (Figure).

Figure 8. Side lengths of special triangles

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

How To

Given trigonometric functions of a special angle, evaluate using side lengths.

1. Use the side lengths shown in (Figure) for the special angle you wish to evaluate.
2. Use the ratio of side lengths appropriate to the function you wish to evaluate.

Evaluating Trigonometric Functions of Special Angles Using Side Lengths

Find the exact value of the trigonometric functions ofusing side lengths.

Try It

Find the exact value of the trigonometric functions ofusing side lengths.

Using Equal Cofunction of Complements

If we look more closely at the relationship between the sine and cosine of the special angles, we notice a pattern. In a right triangle with angles ofandwe see that the sine ofnamelyis also the cosine ofwhile the sine ofnamelyis also the cosine of

Figure 9. The sine ofequals the cosine ofand vice versa.

This result should not be surprising because, as we see from (Figure), the side opposite the angle ofis also the side adjacent tosoandare exactly the same ratio of the same two sides,andSimilarly,andare also the same ratio using the same two sides,and

The interrelationship between the sines and cosines ofandalso holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add toand the right angle isthe remaining two angles must also add up toThat means that a right triangle can be formed with any two angles that add to—in other words, any two complementary angles. So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in (Figure).

Figure 10. Cofunction identity of sine and cosine of complementary angles

Using this identity, we can state without calculating, for instance, that the sine ofequals the cosine ofand that the sine ofequals the cosine ofWe can also state that if, for a given anglethenas well.

Cofunction Identities

The cofunction identities in radians are listed in (Figure).

How To

Given the sine and cosine of an angle, find the sine or cosine of its complement.

1. To find the sine of the complementary angle, find the cosine of the original angle.
2. To find the cosine of the complementary angle, find the sine of the original angle.

Using Cofunction Identities

Iffind

According to the cofunction identities for sine and cosine, we have the following.

Try It

Iffind

Using Trigonometric Functions

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

How To

Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.

1. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
3. Using the value of the trigonometric function and the known side length, solve for the missing side length.

Finding Missing Side Lengths Using Trigonometric Ratios

Find the unknown sides of the triangle in (Figure).

Figure 11.

We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

We rearrange to solve for

We can use the sine to find the hypotenuse.

Again, we rearrange to solve for

Try It

A right triangle has one angle ofand a hypotenuse of 20. Find the unknown sides and angle of the triangle.

missing angle is

Using Right Triangle Trigonometry to Solve Applied Problems

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height.

Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. See (Figure).

Figure 12.

How To

Given a tall object, measure its height indirectly.

1. Make a sketch of the problem situation to keep track of known and unknown information.
2. Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
3. At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
4. Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
5. Solve the equation for the unknown height.

Measuring a Distance Indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle ofbetween a line of sight to the top of the tree and the ground, as shown in (Figure). Find the height of the tree.

Figure 13.

We know that the angle of elevation isand the adjacent side is 30 ft long. The opposite side is the unknown height.

The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent oflettingbe the unknown height.

The tree is approximately 46 feet tall.[/hidden-answer]

Try It

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle ofwith the ground? Round to the nearest foot.

Access these online resources for additional instruction and practice with right triangle trigonometry.

Key Equations

 Trigonometric Functions Reciprocal Trigonometric Functions Cofunction Identities

Key Concepts

• We can define trigonometric functions as ratios of the side lengths of a right triangle. See (Figure).
• The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See (Figure).
• We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See (Figure).
• Any two complementary angles could be the two acute angles of a right triangle.
• If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See (Figure).
• We can use trigonometric functions of an angle to find unknown side lengths.
• Select the trigonometric function representing the ratio of the unknown side to the known side. See (Figure).
• Right-triangle trigonometry facilitates the measurement of inaccessible heights and distances.
• The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See (Figure).

Section Exercises

Verbal

For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

When a right triangle with a hypotenuse of 1 is placed in a circle of radius 1, which sides of the triangle correspond to the x– and y-coordinates?

The tangent of an angle compares which sides of the right triangle?

The tangent of an angle is the ratio of the opposite side to the adjacent side.

What is the relationship between the two acute angles in a right triangle?

Explain the cofunction identity.

For example, the sine of an angle is equal to the cosine of its complement the cosine of an angle is equal to the sine of its complement.

Algebraic

For the following exercises, use cofunctions of complementary angles.

For the following exercises, find the lengths of the missing sides if sideis opposite anglesideis opposite angleand sideis the hypotenuse.

Graphical

For the following exercises, use (Figure) to evaluate each trigonometric function of angle

Figure 14.

For the following exercises, use (Figure) to evaluate each trigonometric function of angle

Figure 15.

For the following exercises, solve for the unknown sides of the given triangle.

Technology

For the following exercises, use a calculator to find the length of each side to four decimal places.

Extensions

Find

Find

Find

Find

A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower isand that the angle of depression to the bottom of the tower isHow tall is the tower?

A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower isand that the angle of depression to the bottom of the tower isHow tall is the tower?

A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument isand that the angle of depression to the bottom of the monument isHow far is the person from the monument?

A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument isand that the angle of depression to the bottom of the monument isHow far is the person from the monument?

There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to beFrom the same location, the angle of elevation to the top of the antenna is measured to beFind the height of the antenna.

There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to beFrom the same location, the angle of elevation to the top of the lightning rod is measured to beFind the height of the lightning rod.

Real-World Applications

A 33-ft ladder leans against a building so that the angle between the ground and the ladder isHow high does the ladder reach up the side of the building?

A 23-ft ladder leans against a building so that the angle between the ground and the ladder isHow high does the ladder reach up the side of the building?

The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to behow far from the base of the tree am I?

Glossary

adjacent side in a right triangle, the side between a given angle and the right angle angle of depression the angle between the horizontal and the line from the object to the observer’s eye, assuming the object is positioned lower than the observer angle of elevation the angle between the horizontal and the line from the object to the observer’s eye, assuming the object is positioned higher than the observer opposite side in a right triangle, the side most distant from a given angle hypotenuse the side of a right triangle opposite the right angle unit circle a circle with a center atand radius 1

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Concepts covered in Mathematics 2 Geometry 9th Standard Maharashtra State Board chapter 8 Trigonometry are Trigonometry, Terms related to right angled triangle, Important Equation in Trigonometry, Method of Using Trigonometric Table, Trigonometric Ratios and Its Reciprocal, Trigonometric Ratios and Its Reciprocal.

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Worked Examples

1. Using Pythagoras’ Theorem
During your GCSE maths exam, you will be required to calculate various mathematical problems using Pythagoras’ theorem:

Example
(a) – The following triangle has the values a = 9 and c = 15. Calculate the value of side ‘b

Solution
(a) – Using Pythagoras’ theorem