# 10.R: Further Applications of Trigonometry (Review) - Mathematics

## 8.1: Non-right Triangles: Law of Sines

For the exercises 1-5 assume (alpha ) is opposite side (a), (eta ) is opposite side (b), and (gamma ) is opposite side (c). Round each answer to the nearest tenth.

1) (eta =50^{circ}, a=105, b=45)

Answer

Not possible

2) (alpha =43.1^{circ}, a=184.2, b=242.8)

3) Solve the triangle.

Answer

(C=120^{circ}, a=23.1, c=34.1)

4) Find the area of the triangle.

Answer

distance of the plane from point (A:2.2) km, elevation of the plane: (1.6) km

## 8.2: Non-right Triangles - Law of Cosines

1) Solve the triangle, rounding to the nearest tenth, assuming (alpha ) is opposite side (a), (eta ) is opposite side (b), and (gamma ) s opposite side (c: a=4, b=6,c=8)$.$

2) Solve the triangle in the Figure below, rounding to the nearest tenth.

Answer

(B=71.0^{circ},C=55.0^{circ},a=12.8)

3) Find the area of a triangle with sides of length (8.3), (6.6), and (9.1).

4) To find the distance between two cities, a satellite calculates the distances and angle shown in the Figure below (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

Answer

(40.6) km

## 8.3: Polar Coordinates

1) Plot the point with polar coordinates (left ( 3,dfrac{pi }{6} ight )).

2) Plot the point with polar coordinates (left ( 5,dfrac{-2pi }{3} ight )).

Answer

3) Convert (left ( 6,dfrac{-3pi }{4} ight )) to rectangular coordinates.

4) Convert (left ( -2,dfrac{3pi }{2} ight )) to rectangular coordinates.

Answer

((0,2))

5) Convert ((7,-2)) to polar coordinates.

6) Convert ((-9,-4)) to polar coordinates.

Answer

((9.8489,203.96^{circ}))

For the exercises 7-9, convert the given Cartesian equation to a polar equation.

7) (x=-2)

8) (x^2+y^2=64)

Answer

(r=8)

9) (x^2+y^2=-2y)

For the exercises 10-11, convert the given polar equation to a Cartesian equation.

10) (r=7cos heta)

Answer

(x^2+y^2=7x)

11) (r=dfrac{-2}{4cos heta +sin heta })

For the exercises 12-13, convert to rectangular form and graph.

12) ( heta =dfrac{3pi }{4})

Answer

(y=-x)

13) (r=5sec heta)

## 8.4: Polar Coordinates - Graphs

For the exercises 1-5, test each equation for symmetry.

1) (r=4+4sin heta)

Answer

symmetric with respect to the line ( heta =dfrac{pi }{2})

2) (r=7)

3) Sketch a graph of the polar equation (r=1-5sin heta). Label the axis intercepts.

Answer

4) Sketch a graph of the polar equation (r=5sin (7 heta )).

5) Sketch a graph of the polar equation (r=3-3cos heta)

Answer

## 8.5: Polar Form of Complex Numbers

For the exercises 1-2, find the absolute value of each complex number.

1) (-2+6i)

2) (4-3i)

Answer

(5)

Write the complex number in polar form.

3) (5+9i)

4) (dfrac{1}{2}-dfrac{sqrt{3}}{2}i)

Answer

(mathrm{cis}left (-dfrac{pi }{3} ight ))

For the exercises 5-6, convert the complex number from polar to rectangular form.

5) (z=5mathrm{cis}left (dfrac{5pi }{6} ight ))

6) (z=3mathrm{cis}(40^{circ}))

Answer

(2.3+1.9i)

For the exercises 7-8, find the product (z_1 z_2) in polar form.

7) (egin{align*} z_1 &= 2mathrm{cis}(89^{circ}) z_2 &= 5mathrm{cis}(23^{circ}) end{align*})

8) (egin{align*} z_1 &= 10mathrm{cis}left ( dfrac{pi }{6} ight ) z_2 &= 6mathrm{cis}left ( dfrac{pi }{3} ight ) end{align*})

Answer

(60mathrm{cis}left ( dfrac{pi }{2} ight ))

For the exercises 9-10, find the quotient (dfrac{z_1}{z_2}) in polar form.

9) (egin{align*} z_1 &= 12mathrm{cis}(55^{circ}) z_2 &= 3mathrm{cis}(18^{circ}) end{align*})

10) (egin{align*} z_1 &= 27mathrm{cis}left ( dfrac{5pi }{3} ight ) z_2 &= 9mathrm{cis}left ( dfrac{pi }{3} ight ) end{align*})

Answer

(3mathrm{cis}left ( dfrac{4pi }{3} ight ))

For the exercises 11-12, find the powers of each complex number in polar form.

11) Find (z^4) when (z=2mathrm{cis}(70^{circ}))

12) Find (z^2) when (z=5mathrm{cis}left ( dfrac{3pi }{4} ight ))

Answer

(25mathrm{cis}left ( dfrac{3pi }{2} ight ))

For the exercises 13-14, evaluate each root.

13) Evaluate the cube root of (z) when

## Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

Heavenly Mathematics traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography Islamic religious rituals celestial navigation polyhedra stereographic projection and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

#### Awards and Recognition

• One of Choice's Outstanding Academic Titles for 2013
• Shortlisted for the 2013 BSHM Neumann Book Prize, British Society for the History of Mathematics

"Once a mainstay of mathematics, spherical trigonometry no longer appears on school curricula. Here, Glen Van Brummelen reasserts the field's importance, sharing in illuminating detail how it figured in astronomy, cartography and our understanding of Earth's rotation."—Rosalind Metcalfe, Nature

"The present book is very well written it leaves a clear impression that the author intended to endear—not merely present and teach—spherical trigonometry to the reader. Although not a history book, there are separate chapters shedding light on the approaches to the subject in the ancient, medieval, and modern times. There are also chapters on spherical geometry, polyhedra, stereographic projection and the art of navigation. The book is thoroughly illustrated and is a pleasant read. Chapters end with exercises the appendices contain a long list of available and not so available textbooks and recommendations for further reading organized by individual chapters. The book made a valuable addition to my library. I freely recommend it to math teachers and curious high schoolers."—Alexander Bogomolny, CTK Insights

"A no-nonsense introduction to spherical trigonometry."Book News, Inc.

"A beautiful popular book."ThatsMaths.com

"Full of academic, textbook content, the book is a delight to math students. So if you are game for a journey into the world of spherical trigonometry, pick up the book. Van Brummelen gives exercises at the end of the chapters that can be fun."—R. Balashankar, Organiser

"Heavenly Mathematicsis a truly enjoyable description of the somewhat forgotten science of spherical trigonometry. . . . As readers discover this discipline, they will also appreciate the beauty inherent in the topic."Choice

"Heavenly Mathematics proves the value of bringing a fascinating piece of mathematical history within the grasp of the general reader."—Florin Diacu, Literary Review of Canada

"Van Brummelen has written a wonderful introduction . . . that draws on the history of [spherical trigonometry] to illuminate the mathematics itself and at the same time gives readers a real sense of what research in the history of early mathematics is all about."Metascience

"[Heavenly Mathematics] is an excellent survey of spherical trigonometry. . . . Simply an appreciation of a beautiful lost subject, with historical overtones. . . . [D]istinguishable for its appealingly fresh style."Mathematical Reviews

"[Heavenly Mathematics] is a lovely book to read. . . . [A] wonderful introduction for anyone who wishes to learn more about this subject. . . . I am in full agreement with the author that spherical trigonometry ought to be brought to a wider audience, and I believe that this is the book to do it."Mathematics Today

"Engaging, clear and not overly technical you can safely lend this book to your friends in the history department. . . . [Heavenly Mathematics] is excellent."Zentralblatt MATH

"Heavenly Mathematics will be of interest to mathematically inclined historians of science and also to students of mathematics and engineering. Because spherical trigonometry is relevant in applications of modern science, this elegant book may even contribute to a renaissance of the subject."—Jan P. Hogendijk, Isis

"This book could serve as an excellent textbook for any secondary school mathematics classroom at or above the level of geometry and certainly trigonometry as the basis for a high school honors class or as a textbook and seminar topic for college students."—Teresa Floyd, Mathematics Teacher

"Any reader of this book (and there should be many) will see how present day mathematics may be viewed through the kaleidoscope of its historical origins. . . . Glen Van Brummelen has written a beautifully produced book that includes fascinating biographical detail at every stage of his narrative."—P.N. Ruane, Mathematical Gazette

"An engaging read that will appeal to historians of science, mathematicians, trigonometry teachers, and anyone interested in the history of mathematics."—Elizabeth Hamm, Aestimatio Critical Reviews in the History of Science

"Heavenly Mathematicsis heavenly, is mathematics, and is so much more: history, astronomy, geography, and navigation, replete with historical illustrations, elegant diagrams, and charming anecdotes. I haven't followed mathematical proofs with such delight in decades. If, as the author laments, spherical trigonometry was in danger of extinction, this book will give it a long-lasting reprieve."—David J. Helfand, president of the American Astronomical Society

"This beautifully written book on an unusual topic, with its wealth of historical information about astronomy, navigation, and mathematics, is greatly to be welcomed."—Robin Wilson, president of the British Society for the History of Mathematics, author of Four Colors Suffice: How the Map Problem Was Solved

"Written by the leading expert on the subject, this engaging book provides an in-depth historical introduction to spherical trigonometry. Heavenly Mathematics breathes new and interesting life into a topic that has been slumbering for far too long."—June Barrow-Green, associate editor of The Princeton Companion to Mathematics

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"Van Brummelen provides not only a wonderful historical treatment of spherical trigonometry but also a modern one that shows how the ancient and medieval methods were replaced by newer and simpler means of problem solving. Many students will find this a fascinating and worthwhile subject."—Victor J. Katz, editor of The Mathematics of Egypt, Mesopotamia, China, India, and Islam

## 10.R: Further Applications of Trigonometry (Review) - Mathematics

Algebra:
Mathematics branch of mathematics that uses basic operations to solve expressions

Linear equations:
A linear equation is an equation involving only the sum or product of constants and the first power of a variable.
where m is the slope and b is the y intercept is the general form of an equation in slope intercept form.
The formula of a slope= where and are two points in ordered pair form.
The general form of point slope form is:
where m is the slope, and is the two points in ordered pair form.

Polynomial equations:
The quadratic formula equals:
where a, b and c are constants of a quadratic equation is of the form
.

Factoring:
Factoring is a process of dividing out a factor from a mathematical expression.
Common factors technique is of the form ax+bx=x(a+b).
Difference of squares: .
Difference of cubes: .
Rationalizing is the process of removing an irrational expression from the numerator of a fraction.

Number line: An axis or ray usually horizontal on which real numbers are represented and ordered from left to right.

Absolute value: For a real number a, it is a if a is greater than or equal to zero or –a if a is less than zero. It is denoted.

Conic sections:
The general equation of a circle:
. Where r is the radius of the circle and is the center of the circle.
The general equation of an ellipse:
.

Natural logarithm: A log taken to base “e”, is approximately 2.7.

This tutorial deals with the algebra concepts needed to build a solid foundation for the development of trigonometric principles. The basic form of linear equations and quadratic equations are discussed in this tutorial to help introduce their applications.

A review of the basic rules of algebra is introduced here. Specific properties of linear equations are shown here with the use of examples. Polynomial equations are also mentioned in this tutorial with the help of examples and graphs.

Specific Tutorial Features:
• The basic representation of linear equations and inequalities are introduced in this tutorial with the help of examples.

Series Features:
• Concept map showing inter-connections of concepts introduced.
• Definition slides introduce terms as they are needed.
• Examples given throughout to illustrate how the concepts apply.
• A concise summary is given at the conclusion of the tutorial.

See all 24 lessons in Trigonometry, including concept tutorials, problem drills and cheat sheets:
Teach Yourself Trigonometry Visually in 24 Hours

## The Mathematics of the Heavens and the Earth: The Early History of Trigonometry

The Mathematics of the Heavens and the Earth is the first major history in English of the origins and early development of trigonometry. Glen Van Brummelen identifies the earliest known trigonometric precursors in ancient Egypt, Babylon, and Greece, and he examines the revolutionary discoveries of Hipparchus, the Greek astronomer believed to have been the first to make systematic use of trigonometry in the second century BC while studying the motions of the stars. The book traces trigonometry’s development into a full-fledged mathematical discipline in India and Islam explores its applications to such areas as geography and seafaring navigation in the European Middle Ages and Renaissance and shows how trigonometry retained its ancient roots at the same time that it became an important part of the foundation of modern mathematics.

The Mathematics of the Heavens and the Earth looks at the controversies as well, including disputes over whether Hipparchus was indeed the father of trigonometry, whether Indian trigonometry is original or derived from the Greeks, and the extent to which Western science is indebted to Islamic trigonometry and astronomy. The book also features extended excerpts of translations of original texts, and detailed yet accessible explanations of the mathematics in them.

No other book on trigonometry offers the historical breadth, analytical depth, and coverage of non-Western mathematics that readers will find in The Mathematics of the Heavens and the Earth.

"Fans of the history of mathematics will be richly rewarded by this exhaustively researched book, which focuses on the early development of trigonometry. . . . Finally, the generous and lucid explanations provided throughout the text make Van Brummelen's history a rewarding one for the mathematical tourist."Mathematics Teacher

"[T]his new and comprehensive history of trigonometry is more than welcome—even more so because it is the first in English. . . . [T]his book will be appreciated by many with an interest—general or more specific—in the history of mathematics."—Steven Wepster, Centaurus

"[T]his book will have wide appeal, for students, researchers, and teachers of history and/or trigonometry. The excerpts selected are balanced and their significances well articulated. . . . It is a book written by an expert after many years of exposure to individual sources and in this way Van Brummelen uniquely advances the field. The book will no doubt become a necessary addition to the libraries of mathematicians and historians alike."—Clemency Montelle and Kathleen M. Clark, Aestimatio

"Van Brummelen's history does far more than simply fill a vacant spot in the historical literature of mathematics. He recounts the history of trigonometry in a way that is both captivating and yet more than satisfying to the crankiest and most demanding of scholars. . . . The Mathematics of the Heavens and the Earth should be a part of every university library's mathematics collection. It's also a book that most mathematicians with an interest in the history of the subject will want to own."—Rob Bradley, MAA Reviews

"I highly recommend the book to all those interested in the way in which the ancient people solve their practical problems and hope that the next volume of this interesting history of spherical and plane trigonometry will appear soon."—Cristina Blaga, Studia Mathematica

"There does not seem to have been a book-length history of trigonometry in English before this fine book. Van Brummelen takes us from the unnamed Egyptians and Babylonians who created trigonometry to the subject's first few centuries in Europe. In between, he deftly traces how it was studied by the astronomers Hipparchus and Ptolemy in classical Greece, and later by a host of scholars in India and the Islamic world."—John H. Conway, coauthor of The Book of Numbers

"This book is the first detailed history of trigonometry in more than half a century, and it far surpasses any earlier attempts. The Mathematics of the Heavens and the Earth is an extremely important contribution to scholarship. It will be the definitive history of trigonometry for years to come. There is nothing like this out there."—Victor J. Katz, professor emeritus, University of the District of Columbia

"A pleasure to read. The Mathematics of the Heavens and the Earth is destined to become the standard reference on the history of trigonometry for the foreseeable future. Although other authors have attempted to tell the story, I know of no other book that has either the breadth or the depth of this one. Van Brummelen is one of the leading experts in the world on this subject."—Dennis Duke, Florida State University

"Van Brummelen presents a history of trigonometry from the earliest times to the end of the sixteenth century. He has produced a work that rises to the highest standards of scholarship but never strays into pedantry. His extensive bibliography cites every work of consequence for the history of trigonometry, copious footnotes and diagrams illuminate the text, and reproductions from old printed works add interest and texture to the narrative."—J. Lennart Berggren, professor emeritus, Simon Fraser University

"This book presents, for the first time in more than a century, a concise history of plane and spherical trigonometry, an important field within applied mathematics. It will appeal to a wide audience thanks to the pleasant style in which it is written, but at the same time it adheres to a very high scholarly standard."—Benno van Dalen, Ludwig Maximilians University, Munich

## Fresno State

Prerequisites: CI 152 AND CI 159 or concurrent enrollment admission to the Single Subject Credential Program or teaching experience. Planning, delivering, and assessing content-specific instruction academic and common core standards identifying specific standards that require literacy strategies. (Instructional materials fee for Single Subject - Art Methods and Materials enrollees, \$10)

Units: 3, Repeatable up to 999 units
Course Typically Offered: Fall, Spring

###### EHD 154B. Final Student Teaching Seminar - Mathematics

Prerequisites: Concurrent enrollment in EHD 155B. Seminar to accompany final student teaching that provides opportunities for candidates to investigate and discuss variety of topics and strategies and to reflect on issues that surface during their student teaching experience.

###### EHD 155B. Student Teaching in Secondary School - Math

Prerequisites: admission to student teaching, EHD 155A, CI 161 (or concurrently, depending on major departmental policy) senior or post baccalaureate standing approval of major department including subject matter competency approval completion of the subject matter preparation program or passing the subject matter examination(s) designated by the California Commission on Teacher Credentialing. Supervised teaching in single subject classroom assignment is for the full day five days per week. CR/NC grading only.

Units: 5-10, Repeatable up to 20 units
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###### MATH 3. College Algebra

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take MATH 3L. Equations and inequalities rectangular coordinates systems of equations and inequalities polynomial, rational, exponential, and logarithmic functions and their graphs complex numbers.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 3L. College Algebra w/ Support

Equations and inequalities rectangular coordinates systems of equations and inequalities polynomial, rational, exponential, and logarithmic functions and their graphs complex numbers.(3 lecture, 3 lab hours)

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Course Typically Offered: Fall, Spring

###### MATH 5. Trigonometry

Prerequisite: Mathematics placement category I or II. Can be taken concurrently with Math 3 for category standing III or IV. Concept of a function, sine and cosine functions, tables and graphs, other trigonometric functions, identities and equations. Trigonometric functions of angles, solution of triangles. (See Duplication of Courses).

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 6. Precalculus

Prerequisite: Mathematics placement category I or II. Basic algebraic properties of real numbers linear and quadratic equations and inequalities functions and graphs polynomials exponential and logarithmic functions analytic trigonometry and functions. G. E. Foundation B4. (3 lecture, 2 activity hours)

Units: 4
Course Typically Offered: Fall, Spring
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###### MATH 10A. Structure and Concepts in Mathematics I

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take Math 10AL. Designed for prospective elementary school teachers. Development of real numbers including integers, rational and irrational numbers, computation, prime numbers and factorizations, and problem-solving strategies. Meets B4 G. E. requirement only for liberal studies majors.

Units: 3
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 10AL. Structure and Concepts in Mathematics I w/Support

Designed for prospective elementary school teachers. Development of real numbers including integers, rational and irrational numbers, computation, prime numbers and factorizations, and problem-solving strategies. Meets B4 G. E. requirement only for liberal studies majors. (3 lecture, 3 lab hours)

Units: 4
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 10B. Structure and Concepts in Mathematics II

Prerequisite: MATH 10A. Designed for prospective elementary school teachers. Counting methods, elementary probability and statistics. Topics in geometry include polygons, congruence and similarity, measurement, geometric transformations, coordinate geometry, and connections between numbers and geometry with selected applications.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 11. Elementary Statistics

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take Math 11L. Illustration of statistical concepts: elementary probability models, sampling, descriptive measures, confidence intervals, testing hypotheses, chi-square, nonparametric methods, regression. It is recommended that students with credit in MATH 75 or MATH 75A and B take MATH 101.

Units: 3
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 11L. Elementary Statistics w/Support

Illustration of statistical concepts: elementary probability models, sampling, descriptive measures, confidence intervals, testing hypotheses, chi-square, nonparametric methods, regression. It is recommended that students with credit in MATH 75 or MATH 75A and B take MATH 101. (3 lecture, 3 lab hours)

###### MATH 45. What Is Mathematics?

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take Math 45L. Covers topics from the following areas: (I) The Mathematics of Social Choice (II) Management Science and Optimization (III) The Mathematics of Growth and Symmetry and (IV) Statistics and Probability. G. E. Foundation B4.

Units: 3
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 45L. What Is Mathematics w/Support

Covers topics from the following areas: (I) The Mathematics of Social Choice (II) Management Science and Optimization (III) The Mathematics of Growth and Symmetry and (IV) Statistics and Probability. G. E. Foundation B4. (3 lecture, 3 lab hours)

###### MATH 70. Calculus for Life Sciences

No credit if taken after MATH 75 or MATH 75A and B. Prerequisite: Mathematics placement category I or II and calculus placement according to department standards. Functions and graphs, limits, derivatives, antiderivatives, differential equations, and partial derivatives with applications in Life Sciences.

Units: 4
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 75. Calculus I

Prerequisites: Mathematics placement category I or II, and calculus placement according to department standards. Functions, graphs, limits, continuity, derivatives and applications, definite and indefinite integrals. (3 lecture, 2 activity hours) G.E. Foundation B4.

Units: 4
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 75A. Calculus with Review IA

Prerequisites: Mathematics placement category I or II, and calculus placement according to department standards. Functions, graphs, limits, continuity, derivatives, and applications, with extensive review of algebra and elementary functions. With MATH 75B, equivalent to MATH 75. G.E. Foundation B4.

Units: 4
Course Typically Offered: Fall, Spring
GE Area: B4

###### MATH 75B. Calculus with Review IB

Prerequisite: MATH 75A. Further applications of derivatives, and definite and indefinite integrals, with extensive review of algebra and elementary functions. With MATH 75A, equivalent to MATH 75.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 76. Calculus II

Prerequisite: MATH 75 or MATH 75A and B. Techniques and applications of integration, improper integrals, conic sections, polar coordinates, infinite series. (3 lecture, 2 activity hours)

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 77. Calculus III

Prerequisite: MATH 76. Vectors, three-dimensional calculus, partial derivatives, multiple integrals, Green's Theorem, Stokes' Theorem.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 81. Elementary Differential Equations with Linear Algebra

Prerequisite: MATH 77. Introduction to ordinary linear differential equations and linear systems of differential equations solutions by Laplace transforms. Solution of linear systems of equations introduction to vector spaces eigenvalues and eigenvectors. Using computer software as an exploratory tool.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 90. Directed Study

Independently arranged course of study in some limited area of mathematics either to remove a deficiency or to investigate a topic in more depth. (1-3 hours, to be arranged)

###### MATH 100. Exploring Mathematics

Prerequisite: MATH 10B. The development of mathematical reasoning, problem solving, and communication skills for effective teaching of mathematics in elementary school.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 101. Statistical Methods

Prerequisite: MATH 70 or MATH 75, or MATH 75A and B no credit if taken after MATH 108. Application of statistical procedures to examples from biology, engineering, and social science one- and two-sample normal theory methods chi-square, analysis of variance, and regression nonparametric methods. Computerized statistical packages are used.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 102. Introduction to Biostatistics

Prerequisites: MATH 6 or MATH 70 or MATH 75 (MATH 70 or MATH 75 may be taken concurrently). Introduction to statistical methods used in biological experiments and data analyses, with emphasis on interpretation, presentation, and writing statistical reports. Topics include describing and visualizing data, sampling methods, probability, experimental design, hypothesis testing, regression, ANOVA and nonparametric tests. This course satisfies the statistics requirement for Biology majors. Students majoring in Biology or a closely related major can benefit greatly from taking this course.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 105. Statistical Programming and Data Analysis

Prerequisite: MATH 11 or MATH 101 or permission of instructor. Introduction to SAS and R through programming and data analysis. Topics include data access, data structure, data management and manipulation, simulations, arrays, matrices, graphics, custom functions, and standard statistical techniques in SAS/R.

Units: 3
Course Typically Offered: Fall

###### MATH 106. Applied Linear Statistical Models

Prerequisites: MATH 75 MATH 11 or MATH 101 or permission of instructor. Topics include simple linear regression, parameter inference, interval estimation, prediction, diagnostics and remedial measures, multiple linear regression, model selection and validation, generalized linear models, ridge regression, LASSO.

Units: 3
Course Typically Offered: Spring

###### MATH 107. Mathematical Statistics

Prerequisite: MATH 77 (may be taken concurrently). Probability theory discrete and continuous distributions random variables conditional distributions multivariate distributions limit theorems maximum likelihood methods.

Units: 3
Course Typically Offered: Fall

###### MATH 108. Advanced Mathematical Statistics

Prerequisite: MATH 107. Statistical inferences sufficiency optimal hypothesis tests inferences from normal theory, nonparametric statistics elementary decision theory Bayesian statistics.

Units: 3
Course Typically Offered: Spring

###### MATH 109. Applied Probability

Prerequisite: MATH 107. Introduction to stochastic processes and their applications in science and industry. Markov chains, queues, stationary time series.

Units: 3
Course Typically Offered: Fall

###### MATH 110. Symbolic Logic

(Similar to PHIL 145 consult department.) Prerequisite: MATH 75 or MATH 75A and B. An informal treatment of the theory of logical inference, statement calculus, truth-tables, predicate calculus, interpretations applications.

Units: 3
Course Typically Offered: Spring

###### MATH 111. Transition to Advanced Mathematics

Prerequisite: MATH 76. Introduction to the language and problems of mathematics. Use of LaTeX as a typesetting tool. Topics include set theory, symbolic logic, types of proofs, and mathematical induction. Emphasis on students constructing, explaining, and justifying mathematical arguments through active learning.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 114. Discrete Structures

Prerequisite: MATH 111. Counting techniques, matrix algebra, graphs, trees and networks, recurrence relations and generating functions, applied modern algebra.

Units: 3
Course Typically Offered: Spring

###### MATH 116. Theory of Numbers

Prerequisite: MATH 111. Divisibility theory in the integers, primes and their distribution, congruence theory, Diophantine equations, number theoretic functions, primitive roots, indices, the quadratic reciprocity law.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 118. Graph Theory

Prerequisite: MATH 111. Trees, connectivity, Euler and Hamilton paths, matchings, chromatic problems, planar graphs, independence, directed graphs, networks.

Units: 3
Course Typically Offered: Spring - even

###### MATH 121. Numerical Analysis I

Prerequisites: MATH 77 and either CSCI 40 or ECE 71. Computer arithmetic, solutions of equations using iterative techniques, interpolation, numerical differentiation, quadrature, and numerical ordinary differential equations. Use of numerical software libraries.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 122. Numerical Analysis II

Prerequisites: MATH 121, MATH 152. Systems of linear equations, Gaussian elimination with pivoting, matrix inversion, determinant of a matrix, SVD, LU and Cholesky factorization of a matrix, iterative techniques, orthogonal matrix, QR factorization, Gram-Schmidt and Householder methods, approximating eigenvalues, systems of nonlinear equations, steepest descent techniques, Newton's method, and rational approximation.

Units: 3
Course Typically Offered: Fall

###### MATH 123. Mathematical Modeling

Prerequisite: MATH 77. Application of mathematical techniques to solve selected problems in areas such as ecology, biology, economics, finance, social sciences, life sciences, physical sciences and engineering. The emphasis will be on building mathematical models and on interpreting the solutions of these models in terms of real-life applications. G.E. IB for BS in Mathematics-Integrated Credential Option only.

Units: 3
Course Typically Offered: Spring - odd
GE Area: IB

###### MATH 128. Applied Complex Analysis

Prerequisite: MATH 77. Analytic functions of a complex variable, contour integration, series, singularities of analytic functions, the residue theorems, conformal mappings emphasis on engineering and physics applications.

Units: 3
Course Typically Offered: Fall

###### MATH 133. Number Theory for Liberal Studies

Prerequisite: MATH 10B or permission of instructor. The historical development of the concept of number and arithmetic algorithms. The magnitude of numbers. Basic number theory. Special numbers and sequences. Number patterns. Modular arithmetic.

Units: 3
Course Typically Offered: Fall

###### MATH 134. Geometry for Liberal Studies

Prerequisite: MATH 10B or permission of instructor. The use of computer technology to study and explore concepts in Euclidean geometry. Topics include, but are not restricted to, properties of polygons, tilings, and polyhedra.

Units: 3
Course Typically Offered: Spring

###### MATH 137. Exploring Statistics

Prerequisite: MATH 10B or permission of instructor. Descriptive and inferential statistics with a focus on applications to mathematics education. Use of technology and activities for student discovery and understanding of data organization, collection, analysis and inference.

Units: 3
Course Typically Offered: Fall

###### MATH 138. Exploring Algebra

Prerequisite: MATH 10B or permission of instructor. Designed for prospective school teachers who wish to develop a deeper conceptual understanding of algebraic themes and ideas needed to become competent and effective mathematics teachers.

Units: 3
Course Typically Offered: Spring

###### MATH 139. Advanced Algebra for Middle School Teachers

Prerequisite: MATH 6 or MATH 138. Basic structures of modern algebra from a middle school mathematics curriculum perspective. Algebraic structures, polynomial equations, and elementary linear algebra.

Units: 4
Course Typically Offered: Fall

###### MATH 143. History of Mathematics

Prerequisite: MATH 75 or MATH 75A and 75B. History of the development of mathematical concepts in algebra, geometry, number theory, analytical geometry, and calculus from ancient times through modern times. Theorems with historical significance will be studied as they relate to the development of modern mathematics.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 145. Problem Solving

Prerequisite: MATH 111 EHD 50 (may be enrolled concurrently). A study of formulation of problems into mathematical form analysis of methods of attack such as specialization, generalization, analogy, induction, recursion, etc. applied to a variety of non-routine problems. Topics will be handled through student presentation.

Units: 3
Course Typically Offered: Fall

###### MATH 149S. Capstone Mathematics for Teachers

Prerequisites: MATH 151 MATH 161 MATH 171 (MATH 161 and 171 may be taken concurrently). Secondary school mathematics from an advanced viewpoint. This course builds on students' work in upper division mathematics to deepen their understanding of the mathematics taught in secondary schools. In collaboration with local in-service teachers and university faculty, students will actively explore topics in number theory, algebra, analysis, geometry, and apply their content knowledge in a service-learning context.

Units: 4
Course Typically Offered: Spring

###### MATH 151. Principles of Algebra

Prerequisite: MATH 111. Equivalence relations groups, cyclic groups, normal sub-groups, and factor groups rings, ideals, and factor rings integral domains and polynomial rings fields and field extensions.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 152. Linear Algebra

Prerequisite: MATH 77. Vector spaces, linear transformations, matrices, determinants, eigenvalues and eigenvectors, linear functions, inner-product spaces, bilinear forms, quadratic forms, orthogonal and unitary transformations, selected applications.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 161. Principles of Geometry

Prerequisite: MATH 111. The classical elliptic, parabolic, and hyperbolic geometries developed on a framework of incidence, order and separation, congruence coordinatization. Theory of parallels for parabolic and hyperbolic geometries. Selected topics of modern Euclidean geometry.

Units: 3
Course Typically Offered: Spring

###### MATH 165. Differential Geometry

Prerequisite: MATH 77 and MATH 111. Study of geometry in Euclidean space by means of calculus, including theory of curves and surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.

Units: 3
Course Typically Offered: Fall

###### MATH 171. Intermediate Mathematical Analysis I

Prerequisite: MATH 111. Natural and rational numbers, real numbers as a complete ordered field, its usual topology, sequences and series of real numbers, functions of a real variable, limits, continuity, uniform continuity, differentiability, generalized mean value theorem, Riemann integrals, and power series.

Units: 4
Course Typically Offered: Fall, Spring

###### MATH 172. Intermediate Mathematical Analysis II

Prerequisite: MATH 77 and MATH 171. Pointwise and uniform convergence of sequences and series of functions, convergence of sequences in higher dimensions, continuity and differentiability of functions of several variables. The inverse and implicit function theorems topics in integration theory in higher dimensions.

Units: 4
Course Typically Offered: Spring

###### MATH 181. Differential Equations

Prerequisite: MATH 81. Definition and classification of differential equations general, particular, and singular solutions existence theorems theory and technique of solving certain differential equations: phase plane analysis, elementary stability theory applications.

Units: 3
Course Typically Offered: Fall

###### MATH 182. Partial Differential Equations

Prerequisites: MATH 81. Classical methods for solving partial differential equations including separation of variables, Green's functions, the Riemann-Volterra method and Cauchy's problem for elliptic, parabolic, and hyperbolic equations applications to theoretical physics.

Units: 3
Course Typically Offered: Spring

###### MATH 190. Independent Study

See Academic Placement. Approved for RP grading.

Units: 1-3, Repeatable up to 6 units
Course Typically Offered: Fall, Spring

###### MATH 191T. Proseminar

Prerequisites: Permission of instructor. Presentation of advanced topics in mathematics in the field of the student's interest.

Units: 1-3, Repeatable up to 9 units

###### MATH 191T. Mathematical Software and Programming

Prerequisite: Math 76 and CSCI 40 or ECE 71. Recommended to be taken concurrently with Math 77 or Math 111. Introduction to programming usage in mathematics and statistics including computer algebra software, interpreted object-oriented high-level programming language, one programming language related to statistics, database management, optimization, and./or cloud computing. This topic may not be repeated for credit. (Offered Fall 2021)

###### MATH 192. Undergraduate Mathematics Seminar

Prerequisite: MATH 76 or consent of the instructor. Presentations on various topics in mathematics. The course is intended for STEM students with a strong interest in mathematics. It is an upper division elective course.

Units: 1, Repeatable up to 3 units
Course Typically Offered: Fall

###### MATH 198. Senior Project

Prerequisites: Senior standing or permission of instructor MATH 151, MATH 171, and MATH 152. Independent investigation and presentation of an advanced topic in mathematics. Satisfies the senior major requirement for the B.A. in Mathematics.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 200. Research Methods in Mathematics Education

This course covers quantitative and qualitative methods of researching how people think and learn about mathematics, and how research informs the teaching of mathematics. Content includes research design, use of statistical analyses, and critical examination of research in mathematics education.

Units: 3
Course Typically Offered: Fall - even

###### MATH 201. Cognition in Mathematics

This course explores theories and empirical studies which examine the development of students' and teachers' knowledge and practices in mathematics. A central theme of the course is the examination of research on the use of technology in the teaching of mathematics.

Units: 3
Course Typically Offered: Fall - odd

###### MATH 202. Fundamental Concepts of Mathematics

Prerequisites: MATH 151, MATH 161 and MATH 171. Fundamental notions regarding number theory, number systems, algebra of number fields functions.

###### MATH 216T. Topics in Number Theory

Prerequisite: MATH 116. An investigation of topics having either historical or current research interest in the field of number theory. (Formerly MATH 216)

Units: 3, Repeatable up to 6 units

###### MATH 220. Coding Theory

Prerequisites: MATH 151 and MATH 152. Basic concepts in coding theory, properties of linear and on-linear codes, standard decoding algorithms, cyclic codes, BCH-codes.

Units: 3
Course Typically Offered: Spring - even

###### MATH 223. Applied Operator Theory

Prerequisite: graduate standing or permission of instructor. Fundamentals of abstract spaces and spectral theory of operators with applications. Resolvent set and spectrum of a linear operator. Bounded and unbounded linear operators. Compact operators and the Fredholm alternative. Symmetry and self-adjointness.

Units: 3
Course Typically Offered: Spring - odd

###### MATH 228. Functions of a Complex Variable

Prerequisite: MATH 128. Representation theorems of Weierstrass and Mittag-Leffler, normal families, conformal mapping and Riemann mapping theorem, analytic continuation, Dirichlet problem.

Units: 3
Course Typically Offered: Spring - odd

###### MATH 232. Mathematical Models with Technology

Prerequisite: graduate standing in mathematics or permission of instructor. A technology-assisted study of the mathematics used to model phenomena in statistics, natural science, and engineering.

Units: 3
Course Typically Offered: Fall - even

###### MATH 250. Perspectives in Algebra

Prerequisite: graduate standing in mathematics or permission of instructor. Study of advanced topics in algebra, providing a higher perspective to concepts in the high school curriculum. Topics selected from, but not limited to, groups, rings, fields, and vector spaces.

###### MATH 251. Abstract Algebra I

Prerequisite: MATH 151 or permission of instructor. Semi-direct products of groups isomorphism theorems. Group actions Sylow theorems classification of groups of small order finitely generated Abelian groups. Rings and ideals quotient rings domains (ED, PID, UFD) polynomial rings.

Units: 3
Course Typically Offered: Spring

###### MATH 252. Abstract Algebra II

Prerequisite: MATH 251. Field extensions automorphisms of fields Galois theory. Additional topics to be chosen from (1) modules, (2) linear and multilinear algebra and (3) representation theory.

Units: 3
Course Typically Offered: Fall - odd

###### MATH 260. Perspectives in Geometry

Prerequisite: MATH 151 and MATH 152 or permission of instructor. Geometry from a transformations point of view. Projective geometry: theorems of Ceva, Menelaus, Desargues, and Pappus conics coordinatization. Transformations of the plane (Euclidean and projective) tessellations wallpaper groups. Further topics to be selected from Incidence Geometry, Differential Geometry, or Algebraic Geometry.

Units: 3
Course Typically Offered: Spring - even

###### MATH 263. Point Set Topology

Prerequisite: MATH 172. Basic concepts of point set topology, set theory, topological spaces, continuous functions connectivity, compactness and separation properties of spaces. Topics selected from function spaces, metrization, dimension theory.

Units: 3
Course Typically Offered: Fall - even

###### MATH 270. Perspectives in Analysis

Prerequisite: graduate standing in mathematics or permission of instructor. An overview of the development of mathematical analysis, both real and complex. Emphasizes interrelation of the various areas of study , the use of technology, and relevance to the high school mathematics curriculum.

###### MATH 271. Real Analysis

Prerequisite: MATH 172. Lebesgue's measure and integration theory on the real line. Limit theorems and types of convergence. Lp spaces. Differentiation and integration.

Units: 3
Course Typically Offered: Fall

###### MATH 272. Functional Analysis

Prerequisite: MATH 271 or permission of instructor. Elements of the theory of abstract spaces. The three fundamental principles of linear functional analysis (Hahn-Banach Theorem, Uniform Boundedness Principle, and Open Mapping Theorem) and their implications. Duality and reflexivity of normed vector spaces, geometry of Hilbert spaces. (Formerly MATH 291T)

Units: 3
Course Typically Offered: Spring - even

###### MATH 290. Independent Study

See Academic Placement. Approved for RP grading.

Units: 1-3, Repeatable up to 6 units
Course Typically Offered: Fall, Spring

###### MATH 291T. Seminar

Prerequisite: graduate standing. Presentation of current mathematical research in field of student's interest.

Units: 1-3, Repeatable up to 6 units

###### MATH 298. Research Project in Mathematics

Prerequisite: graduate standing. Independent investigation of advanced character as the culminating requirement for the master's degree. Approved for RP grading.

Units: 3
Course Typically Offered: Fall, Spring

###### MATH 298C. Project Continuation

Prerequisite: Project MATH 298. For continuous enrollment while completing the project. May enroll twice with department approval. Additional enrollments must be approved by the Dean of Graduate Studies.

Units: 0
Course Typically Offered: Fall, Spring

###### MATH 299. Thesis in Mathematics

Prerequisite: See Criteria for Thesis and Project. Preparation, completion, and submission of an acceptable thesis for the master's degree. Approved for RP grading.

Reduce the following to one trigonometric ratio:

(cos^2 heta an^2 heta + an^2 heta sin^2 heta)

(1-sin heta cos heta an heta)

Prove the following identities and state restrictions where appropriate:

Restrictions: undefined where (cos heta = ext<0>, sin heta = ext<1>) and where ( an heta)is undefined.

(^<2>alpha + left(cosalpha - analpha ight)left(cosalpha + analpha ight)=1-< an>^<2>alpha)

Restrictions: undefined where ( an heta) is undefined.

Restrictions: undefined where (cos heta = ext<0>) and where ( an heta)is undefined.

Trigonometry Lessons Part 1: Definitions

By Jane a retired Math instructor who taught at the University of Massachusetts Lowell

This video defines and provides examples of the six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

How to Remember the Unit Circle (mathbff)

How to Remember the Unit Circle (mathbff)
MIT grad shows how to remember the unit circle angles and points. The unit circle is a circle of radius one. There are patterns within the unit circle that make it easier to understand and to memorize.

## 10.R: Further Applications of Trigonometry (Review) - Mathematics

Algebra:
Mathematics branch of mathematics that uses basic operations to solve expressions
Linear equations:
A linear equation is an equation involving only the sum or product of constants and the first power of a variable.
where m is the slope and b is the y intercept is the general form of an equation in slope intercept form.
The formula of a slope= where and are two points in ordered pair form.
The general form of point slope form is:
where m is the slope, and is the two points in ordered pair form.
Polynomial equations:
The quadratic formula equals:
where a, b and c are constants of a quadratic equation is of the form
.
Factoring:
Factoring is a process of dividing out a factor from a mathematical expression.
Common factors technique is of the form ax+bx=x(a+b).
Difference of squares: .
Difference of cubes: .
Rationalizing is the process of removing an irrational expression from the numerator of a fraction.
Order and Inequalities:
Number line: An axis or ray usually horizontal on which real numbers are represented and ordered from left to right.
Absolute value: For a real number a, it is a if a is greater than or equal to zero or –a if a is less than zero. It is denoted.
Conic sections:
The general equation of a circle:
. Where r is the radius of the circle and is the center of the circle.
The general equation of an ellipse:
.

Natural logarithm: A log taken to base “e”, is approximately 2.7.
Natural Log Function: .
Exponential function: .

This tutorial deals with the algebra concepts needed to build a solid foundation for the development of trigonometric principles. The basic form of linear equations and quadratic equations are discussed in this tutorial to help introduce their applications.

A review of the basic rules of algebra is introduced here. Specific properties of linear equations are shown here with the use of examples. Polynomial equations are also mentioned in this tutorial with the help of examples and graphs.

Specific Tutorial Features:
• The basic representation of linear equations and inequalities are introduced in this tutorial with the help of examples.

Series Features:
• Concept map showing inter-connections of concepts introduced.
• Definition slides introduce terms as they are needed.
• Examples given throughout to illustrate how the concepts apply.
• A concise summary is given at the conclusion of the tutorial.

See all 24 lessons in Trigonometry, including concept tutorials, problem drills and cheat sheets:
Teach Yourself Trigonometry Visually in 24 Hours

## Trigonometry in Civil Engineering

The writers from SBE Builders explain that although trigonometry originates from the Greek words “trigōnon” and “metron,” meaning triangle and measurement respectively, a less-emphasized aspect of trigonometry is that it is a method of calculating the x and y values of a point on the perimeter of a circle from its center.

Many jobs use trigonometry. Writers from Reference.com point out that civil engineers are among the many professions that use trigonometry on a daily basis. Civil engineering is an important part of the construction process, with civil engineers designing structures before they are built. Civil engineers interface with the construction companies and contractors who carry out the builds of their designs.

Whether it’s determining how many braces they need to support a bridge or planning how steeply to design a road along, around or through a hill, both engineers and construction teams are very focused on mathematics of their projects, including trigonometry.

## Trigonometry Frequently Asked Questions

What do you Mean by Trigonometry?
Ans. – Trigonometry is one of the branches of mathematics which deals with the relationship between the sides of a triangle (right triangle) with its angles. There are 6 trigonometric functions to define It.

What are the Different Trigonometric Functions?
Ans. – The 6 trigonometric functions are: Sine function, Cosine function, Tan function, Sec function, Cot function, Cosec function

Who is the Father of Trigonometry?
Ans. – Hipparchus was a Greek astronomer who lived between 190-120 B.C. He is considered the father of trigonometry.

What are the Applications of Trigonometry in Real Life?
Ans. – The real life applications of trigonometry is in the calculation of height and distance. Some of the sectors where the concepts of trigonometry is extensively used are aviation department, navigation, criminology, marine biology, etc.