6.E: Periodic Functions (Exercises) - Mathematics

6.1: Graphs of the Sine and Cosine Functions

In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions


1) Why are the sine and cosine functions called periodic functions?


The sine and cosine functions have the property that (f(x+P)=f(x)) for a certain (P). This means that the function values repeat for every (P) units on the (x)-axis.

2) How does the graph of (y=sin x) compare with the graph of (y=cos x)? Explain how you could horizontally translate the graph of (y=sin x) to obtain (y=cos x).

3) For the equation (A cos(Bx+C)+D), what constants affect the range of the function and how do they affect the range?


The absolute value of the constant (A) (amplitude) increases the total range and the constant (D) (vertical shift) shifts the graph vertically.

4) How does the range of a translated sine function relate to the equation (y=A sin(Bx+C)+D)?

5) How can the unit circle be used to construct the graph of (f(t)=sin t)?


At the point where the terminal side of (t) intersects the unit circle, you can determine that the (sin t) equals the (y)-coordinate of the point.


For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum (y)-values and their corresponding (x)-values on one period for (x>0). Round answers to two decimal places if necessary.

6) (f(x)=2sin x)

7) (f(x)=dfrac{2}{3}cos x)


amplitude: (dfrac{2}{3});period: (2pi );midline: (y=0);maximum: (y=dfrac{2}{3}) occurs at (x=0);minimum: (y=-dfrac{2}{3}) occurs at (x=pi );for one period, the graph starts at (0) and ends at (2pi ).

8) (f(x)=-3sin x)

9) (f(x)=4sin x)


amplitude: (4); period: (2pi );midline: (y=0);maximum (y=4) occurs at (x=dfrac{pi }{2});minimum: (y=-4) occurs at (x=dfrac{3pi }{2});one full period occurs from (x=0) to (x=2pi)

10) (f(x)=2cos x)

11) (f(x)=cos (2x))


amplitude: (1); period: (pi);midline:

Periodic Functions

Periodic functions are applied to study signals and waves in electrical and electronic systems, vibrations in mechanical and civil engineering systems, waves in physics and wireless systems and has many other applications.
The graph of a periodic function repeats itself over cycles for ( x ) in the domain of the function. If ( f ) is known over one cycle, it is known everywhere over the domain of ( f ) since the graph repeats itself.
A function ( f ) is periodic with period ( P ) if
( f(x) = f(x + P) ) , for ( x ) in the domain of ( f ).
( P ) is the smallest positive real number for which the above condition holds. In the graph below is shown a periodic function with two cycles as an example. The period (P) is the distance, along the x axis, between any two points making a cycle as shown in the graph below.
( P = x_2 - x_1 = x_4 - x_3 )

Example 1
All six trigonometric functions are periodic.

    ( sin(x + 2pi ) = sin(x) ) , the period of ( sin(x) ) is equal to ( P = 2pi )
    The graph of ( sin(x) ) is shown below with one cycle, in red, whose length over the x axis is equal to one period P given by: ( P = 2 pi - 0 = 2 pi )

Even & Odd Functions and Periodic Function

The graph of an even function is symmetric about the y−axis (i.e. if (x , y) lies on the curve, then (− x , y) also lies on the curve) and that of an odd function is symmetric about the origin (i.e. if (x , y) lies on the curve , then (− x , − y) also lies on the curve).


⋄ Some times it is easy to prove that f(x) − f(−x) = 0 for even function and f(x) + f(−x) = 0 for odd functions .

⋄ A function can be even or odd or neither even nor odd.

⋄ Every function defined in symmetric interval D

( i.e. x∈ D ⇒ −x ∈ D) can be expressed as a sum of an even and an odd function.

It can now easily be shown that h(x) is even and g(x) is odd.

⋄ The first derivative of an even function is an odd function and vice versa. This is left as an exercise for you to prove.

⋄ If x = 0 ∈ domain of f, then for odd function f(x) which is continuous at x = 0 , f(0) = 0 i.e. if for a function, f(0) ≠ 0, then that function can not be odd.

It follows that for a differentiable even function f ‘(0) = 0 i.e if for a differentiable function f ‘(0) ≠ 0 then the function f cannot be even

Illustration : Which of the following functions is (are) even , odd or neither:

Solution: (i) $large f(-x) = (-x)^2 sin(-x) $

(v) f(-x) = sin(-x) – cos(-x) = -sinx – cos x

Hence f(x) is neither even nor odd.

Extension of Domain:

Let a function be defined on certain domain which is entirely non-negative (or non positive). The domain of f(x) can be extended to the

set X = <−x : x∈ domain of f(x)>in two ways:

(i) Even extension: The even extension is obtained by defining a new function f(−x) for x ∈ X , such that f(−x) = f(x).

(ii) Odd extension: The odd extension is obtained by defining a new function f(-x) for x∈X, such that f(−x) = − f(x)

Illustration :

If $ displaystyle f(x) = left<egin x^3 + x^2 & mathrm 0 le x leq 2 x+2 & mathrm 2 < x le 4end ight. $

Then find the even and odd extension of f(x).

Solution: The even extension of f(x) is as follows:

$ displaystyle g(x) = left<egin -x + 2 & mathrm<,> -4 le x < -2 -x^3 + x^2 & mathrm <,> -2 < x le 0 end ight. $

The odd extension of f(x) is as follows:

$ displaystyle h(x) = left<egin x – 2 & mathrm<,> -4 le x < -2 x^3 – x^2 & mathrm <,> -2 < x le 0 end ight. $

Periodic Function:

A function f: X →Y is said to be a periodic function if there exists a positive real number p such that :

The least of all such positive numbers p is called the principal period or simply period of f.

All periodic functions can be analyzed over an interval of one period within the domain as the same pattern shall be repetitive over the entire domain.

sinx , cosx, secx are periodic functions with period 2π

tanx, cotx are periodic with period π

f(x) is periodic with period 1

(can you prove it mathematically?).

There are two types of questions asked in the examination. You may be asked to test for periodicity of the function or to find the period of the function.

In the former case you just need to show that f(x + T) = f(x) for same T ( > 0) independent of x whereas in the latter, you are required to find a least positive number T independent of x for which f(x + T)= f(x) is satisfied.

The following points are to be remembered:

If f(x) is periodic with period p, then af(x) +b where a , b ∈ R (a ≠ 0) is also periodic with period p.

∎ If f(x) is periodic with period p , then f(ax + b) where a, b ∈ R ( a ≠0) is also period with period $large frac

<|a|>$ .

∎ Let f(x) has period p = m/n (m, n ∈ N and co-prime) and g(x) has period q = r/s (r, s ∈ N and co-prime) and let t be the LCM of p and q i.e. $large t = frac$

Then t shall be the period of f + g provided there does not exist a positive number k (< t) for which

f(k + x) + g(k + x) = f(x) + g(x) , else k will be the period.

The same rule is applicable for any other algebraic combination of f(x) and g(x)


LCM of p and q always exist if p/q is a rational quantity. If p/q is irrational then algebraic combination of f and g is non-periodic.

∎ sin n x, cos n x, cosec n x and sec n x have period 2π if n is odd and π if n is even.

∎ tan n x and cot n x have period π whether n is odd or even.

∎ A constant function is periodic but does not have a well-defined period.

∎ If g is periodic then fog will always be a periodic function. Period of fog may or may not be the period of g.

∎ If f is periodic and g is strictly monotonic (other than linear) then fog is non-periodic.

Illustration : Find the periods (if periodic) of the following functions, where [.] denotes the greatest integer function.

6.E: Periodic Functions (Exercises) - Mathematics

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Bibliography & References:

- Běloun F. a kolektiv: Sbírka úloh z matematiky pro základní školy SPN Praha, 1985

- Bálint Ľ., Bobok J., Križalkovičová M., Lukátšová J.: Zbierka úloh z matematiky na prijímacie skúšky na stredné školy SPN Bratislava, 1986

- Richtáriková S., Kyselová D.: Matematika Enigma Nitra, 1999

- Hudcová M., Kubičíková L.: Sbírka úloh z matematiky pro SOŠ, SOU a nástavbové studium Prometheus Praha, 2010

- Bača M. a kolektív: Zbierka riešených a neriešených úloh z matematiky Technická univerzita v Košiciach Košice, 2011

- Peller F., Starečková A., Pinda Ľ.: Matematika Ekonóm Bratislava, 2009

- Heřmánek L. a kolektív: Sbírka příkladů z matematiky I ve strukturovaném studiu VŠCHT Praha, 2005

- Blaško R.: Matematická analýza I Žilinská univerzita Žilina, 2009

- Tesař J.: Sbírka úloh z matematiky pro fyziky Jihočeská univerzita v Českých Budějovicích České Budějovice

- Eliaš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky Alfa Bratislava, 1966

- Štědrý M., Krylová N.: Sbírka příklad ů z matematiky I PřF UK Praha, 1994

- Hošková Š., Kuben J., Račková P.: Integrální počet funkcí jedné proměnné Vysoká škola báňská Technická univerzita Ostrava, 2006

- Slavík V., Dvořáková Š.: Integrální počet Česká zemědělská univerzita v Praze, 2007

- Oršanský P.: Základy matematickej štatistiky Prešovská univerzita v Prešove, 2009

Practice Test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

f ( x ) = 5 sin ( 3 ( x − π 6 ) ) + 4 f ( x ) = 5 sin ( 3 ( x − π 6 ) ) + 4

f ( x ) = 3 cos ( 1 3 x − 5 π 6 ) f ( x ) = 3 cos ( 1 3 x − 5 π 6 )

f ( x ) = − 2 tan ( x − 7 π 6 ) + 2 f ( x ) = − 2 tan ( x − 7 π 6 ) + 2

For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

Give in terms of a sine function.

Give in terms of a sine function.

Give in terms of a tangent function.

For the following exercises, find the amplitude, period, phase shift, and midline.

y = 8 sin ( 7 π 6 x + 7 π 2 ) + 6 y = 8 sin ( 7 π 6 x + 7 π 2 ) + 6

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t t is the number of hours since midnight, find a function for the temperature, D , D , in terms of t . t .

Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

For the following exercises, find the period and horizontal shift of each function.

n ( x ) = 4 csc ( 5 π 3 x − 20 π 3 ) n ( x ) = 4 csc ( 5 π 3 x − 20 π 3 )

Write the equation for the graph in Figure 1 in terms of the secant function and give the period and phase shift.

For the following exercises, graph the functions on the specified window and answer the questions.

For the following exercises, let f ( x ) = 3 5 cos ( 6 x ) . f ( x ) = 3 5 cos ( 6 x ) .

What is the largest possible value for f ( x ) ? f ( x ) ?

What is the smallest possible value for f ( x ) ? f ( x ) ?

Where is the function increasing on the interval [ 0 , 2 π ] ? [ 0 , 2 π ] ?

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

f ( x ) = 5 cos ( 3 x ) + 4 sin ( 2 x ) f ( x ) = 5 cos ( 3 x ) + 4 sin ( 2 x )

For the following exercises, find the exact value.

For the following exercises, suppose sin t = x x + 1 . sin t = x x + 1 . Evaluate the following expressions.

For the following exercises, determine whether the equation is true or false.

arcsin ( sin ( 5 π 6 ) ) = 5 π 6 arcsin ( sin ( 5 π 6 ) ) = 5 π 6

arccos ( cos ( 5 π 6 ) ) = 5 π 6 arccos ( cos ( 5 π 6 ) ) = 5 π 6

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.

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    2 Answers 2

    $f(x)=sum_ frac<1> e^<>>$ is almost periodic as the uniform limit of trig polynomials.

    Assume now by contradiction that $F(x)=int_0^x f(t) dt$ is almost periodic. Then the Fourier Bohr coefficient at $e^$ is given by $b_>=langle F(x), e^<-ifrac> angle =lim_ frac<1>int_<0>^T F(x) e^<-ifrac< x>>dx$

    Use integration by parts to conclude that $b_>=frac<1>$. But this contradicts the Bessel inequality $sum_ |b_>|^2 leq langle F(x), F(x) angle $

    P.S. If you need any details for any step let me know.

    Now, since we assumed that $F(x)$ is bounded, we have $lim_ frac<1> left( F(x) frac<-i>e^<-ifrac< x>>|_0^T ight)=0$

    $AP$ is a Banach space with the uniform norm, being precisely the uniformly closed span of a set of functions. It turns out that $AP_0$, the space of functions in $AP$ with mean zero, is a closed subspace. (It's not hard to show this directly (details below). Or if you want to be cool, you get into the not-quite-trivial theory: In fact $fin AP$ has mean zero if and only if $f$ is in the closed span of the functions $e_a(t)=e^$ for $a e0$.)

    If $Tf$ were bounded for every $fin AP_0$ the Closed Graph Theorem would show that $||Tf||_inftyle c||f||_infty$ for $fin AP_0$, which is clearly not so.

    Details: Define $Af(x)=frac1xint_0^x f(t),dt$. Say $f_nin AP_0$ and $f_n o0$ uniformly. Then $Af_nin C_0([0,infty))$ and $Af_n o Af$ uniformly, so $Afin C_0$, hence $Mf=0$.

    Note Usually the mean value is defined as $limfrac1<2T>int_<-T>^T$. Wasn't clear to me immediately, but that's actually equal to the mean value defined above. Proof: The two are the same if $f(t)=e^$.

    All Together Now!

    We can have all of them in one equation:

    • amplitude is A
    • period is 2 π /B
    • phase shift is C (positive is to the left)
    • vertical shift is D

    And here is how it looks on a graph:

    Note that we are using radians here, not degrees, and there are 2 π radians in a full rotation.

    Example: sin(x)

    This is the basic unchanged sine formula. A = 1, B = 1, C = 0 and D = 0

    So amplitude is 1, period is 2 π , there is no phase shift or vertical shift:

    Example: 2 sin(4(x − 0.5)) + 3

    • amplitude A = 2
    • period 2 π /B = 2 π /4 = π /2
    • phase shift = −0.5 (or 0.5 to the right)
    • vertical shift D = 3
    • the 2 tells us it will be 2 times taller than usual, so Amplitude = 2
    • the usual period is 2 π , but in our case that is "sped up" (made shorter) by the 4 in 4x, so Period = π /2
    • and the −0.5 means it will be shifted to the right by 0.5
    • lastly the +3 tells us the center line is y = +3, so Vertical Shift = 3

    Instead of x we can have t (for time) or maybe other variables:

    Example: 3 sin(100t + 1)

    First we need brackets around the (t+1), so we can start by dividing the 1 by 100:

    3 sin(100t + 1) = 3 sin(100(t + 0.01))

    • amplitude is A = 3
    • period is 2 π /100 = 0.02 π
    • phase shift is C=0.01 (to the left)
    • vertical shift is D = 0


    • [ Fourier analysis on finite abelian groups ] . [ updated 17:10, Oct 17, 2007] . Decomposition of the regular representation of a finite abelian group, that is, acting on functions on itself, under translation. Assumes only spectral theory on finite-dimensional complex vectorspaces.
    • [14] (DRAFT) [ Dirichlet series from automorphic forms] . [ updated 11:10, Oct 23, 2018] . Beginning of study of Dirichlet series with meromorphic continuation and functional equation obtained from automorphic forms, both holomorphic ones and waveforms.
    • [13] (DRAFT) [ toward waveforms] . [ updated 11:10, Oct 23, 2018] . Beginning of study of eigenfunctions for the invariant Laplacian on the upper half-plane. Introduction of (non-holomorphic) Eisenstein series, cuspforms.
    • [12] (DRAFT) [ Invariant differential operators] . [ updated 14:10, Oct 28, 2010] . More intrinsic discussion of differential operators related to group actions. Introduction of Casimir operator in the universal enveloping algebra attached to a Lie algebra, etc. No assumption of prior acquaintance with Lie algebras or Lie groups .
    • [11] (DRAFT) [ Functions on spheres] . [ updated 15:10, Oct 10, 2010] . Harmonic polynomials, Fourier-Laplace series, Sobolev spaces, on spheres. Duals, distributions (generalized functions).
    • [10] (Functions on the line)
      • [exercises 10]. [ updated 13:12, Dec 11, 2005] . Easy exercises about distributions, Fourier transforms, tempered distributions
      • [exercises 09] . [ updated 13:12, Dec 11, 2005]
      • [exercises 07]. [ updated 10:12, Dec 08, 2005]
      • [Surjectivity of SL(2,Z)-> SL(2,Z/p) ] . [ updated 12:12, Dec 08, 2005]
      • [exercises 06]. [ updated 13:11, Nov 19, 2005] . Constructions of periodic functions, orbits on projective spaces, some counting issues, other oddments.
      • [exercises 05]. [pdf] . basic classical viewpoint on p-adic numbers
      • [exercises 04]. [ updated 14:10, Oct 30, 2005] . metrics, completeness, more colimits, more automorphism of solenoids
      • [exercises 03]. [ updated 14:10, Oct 30, 2005] . Some topology, some commutation of operators, some galois theory.
      • [exercises 02]. [ updated 17:10, Oct 03, 2005] . Further mapping-property exercises.
      • [exercises 01]. . Some basic mapping-property exercises.

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      Table of Contents

      In this text, the reader will learn that all the basic functions that arise in calculus&mdashsuch as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet&mdashare naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here.

      This book could serve for either a one-semester course or a two-semester course in complex analysis for beginning graduate students or for well-prepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus.

      In this Chapter

      Functions Overview

      1. Introduction to Functions - definition of a function, function notation and examples

      2. Functions from Verbal Statements - turning word problems into functions

      Graphs of Functions

      3. Rectangular Coordinates - the system we use to graph our functions

      4. The Graph of a Function - examples and an application

      Domain and Range of a Function - the `x`- and `y`-values that a function can take

      5. Graphing Using a Computer Algebra System - some thoughts on using computers to graph functions

      6. Graphs of Functions Defined by Tables of Data - often we don't have an algebraic expression for a function, just tables

      7. Continuous and Discontinuous Functions - the difference becomes important in later mathematics

      8. Split Functions - these have different expressions for different values of the independent variable

      9. Even and Odd Functions - these are useful in more advanced mathematics


      Exercise 1

      • Make a unit circle through two points (A=(0,0)) and (B=(1,0)).
      • Enter a point (C) on the circle and mark the angle (angle BAC).
      • Make sure that radians is chosen under the Advanced -tab under Options->Settings. .
      • Use the tool Circular Arc on the points (A, B, C).
      • Change the radius of the circle and the angle. Observe the length of the arc.

      What is the length of the arc in terms of the radius (r) and the angle (alpha), if radians are used as angle unit?

      What is the length if degrees are used as angle unit?

      Exercise 2

      • Delete the arc.
      • Use the tool Circular Sector on the points (A, B, C).
      • Change the radius of the circle and the angle. Observe the area of the sector.

      What is the area of the sector in terms of the radius (r) and the angle (alpha), if radians are used as angle unit?

      What is the area if degrees are used as angle unit?

      by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License