# Symmetry (Odd & Even) and Periodicity of Trigonometric Function

• Form a unit circle.

• Find the symmetry (odd and even) of the given function by using the unit circle.

• Find the periodicity of the given function by using the unit circle.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

• Form a unit circle.

• Find the symmetry (odd and even) of the given function by using the unit circle.

• Find the periodicity of the given function by using the unit circle.

Use the Unit Circle to Explain Symmetry (Odd and Even) and Periodicity of Trigonometric Functions

Objectives

Form a unit circle.

Find the symmetry (odd and even) of the given function by using the unit circle.

Find the periodicity of the given function by using the unit circle.

Vocabulary

Function: A relation between a set of inputs and a set of outputs with each input is related to exactly one output.

Unit Circle: A unit circle is a circle with a radius of one.

Radians: The radian is the standard unit of angular measure.

Symmetry: An action happening at regularly spaced periods of time.

Periodic Function: A function that repeats its values in regular intervals or periods.

Introduction

The unit circle is a way to remember trigonometric values. We are going to learn how to remember the values of trigonometric functions using the unit circle and how to use the unit circle to find the symmetry of the function and periodicity of trigonometric functions.

Unit Circle

A unit circle is a circle with a radius of one (a unit radius).

In trigonometry, the unit circle is centered at the origin.

The radians measures of the unit circle is shown below:

The unit circle below shows the corresponding values (cos, sin) for the the radian measures.

We can use unit to derive basic identities.

The point (x, y) is in the first quadrant, and the lengths x and y become the legs of the right triangle whose hypotenuse is 1.

By the Pythagorean theorem, we have x^2+ y^2=1.

If we examine angle ϴ in this unit circle, we can see that

cos ϴ = x/1=x and sin ϴ = y/1=y

Which show us that in a unit circle, cos ϴ = x and sin ϴ = y

So (x, y) = (cos ϴ, sin ϴ)

Sine is represented by the vertical leg.

Cosine is represented by the horizontal leg.

Note that x^2+ y^2=r^2=1 becomes 〖cos^2 ϴ〗+ 〖sin^2 ϴ〗〖=1 〗

Symmetry (Odd and Even Functions)

For any real number t, the points p(t) and p(-t) on the unit circle are located on the terminal side of an angle of t and -t radians, respectively.

If f(t) = f(-t), then the function is an even function

If f(t) = - f(t), then the function is an odd function.

Look at the following examples:

Example 1

Consider f(t) = sin t. Check whether it is odd or even.

f(-t) = sin (-t) = - sin t = - f( t)

That is, f(t) = - f( t).

Therefore, sin t is an odd function.

Similarly, cosec t, tan t, and cot t are odd functions.

Example 2

Consider f(t) = cos t. Check whether it is odd or even.

f(-t) = cos (-t) = cos t = f(t)

Therefore, cos t is an even function.

Similarly. sec t is an even function.

Remember

The even-odd identities are as follows:

sin(-x) = -sin(x) Odd

cos(-x) = cos(x) Even

csc(-x) = -csc(x) Odd

sec(-x) = sec(x) Even

tan(-x) = -tan(x) Odd

cot(-x) = -cot(x) Odd

Example 3

Find the exact values of sin t, cos t, tan t, sec t, cosec t, cot t for the real number t = -π/4. Check whether it is odd or even.

a) sin t = sin (-π/4) = - sin π/4

By using the unit circle, the value of sin π/4 = √2/2

Therefore, - sin π/4 = - √2/2. This is an odd function.

b) cos t = cos (-π/4) = cos π/4 By using the unit circle, the value of cos π/4 = √2/2. This is an even function.

c) tan (- t) = sin〖(-t)/cos〖(-t)〗 〗=((-√2/2))/(√2/2) = -1

This is an odd function.

d) sec (- t) = (1/〖cos (-t)〗█()=(1/(√2/2))=2/√[email protected] )

This is an even function.

e) cosec (- t) = 1 / sin (-t) = (1/(-√2/2))=(-2/√2)

This is an odd function.

f) cot (- t)= 1 / tan (- t) = (1/-1) = -1

This is an odd function.

Periodicity of Trigonometric Functions

A periodic function is a function that repeats its values after some definite period has been added to its independent variable.

For example, consider the hands of a clock. Let time be the independent variable. The position of the minute hand repeats every 60 minutes.

Definition

A function f is said to be a periodic function if there exists a positive number p such that f (t) = f (t + p) for all t in the domain of f. The smallest possible value of p is called the period of the function.

The trigonometric functions repeat their values over and over in a regular pattern. So they are periodic functions.

The circumference of a unit circle is 2π.

The smallest value of p for which the sine and cosine functions repeat is 2π.

So the period of sine and cosine functions are 2π.

Similarly, the smallest value of p for which the tan and cot functions repeat is π.

So the period of tan and cot functions are π.

The following are the general periodic functions:

sin (t + 2πk) = sin (t)

cos (t + 2πk) = cos (t)

tan (t + πk) = tan (t)

cot (t + πk) = cot (t), where k = any integer

Example 4

Evaluate sin (7π/3) by using the unit circle.

We can rewritten sin (7π/3) as sin ((π/3)+ 2π*1)

We know that sin (t + 2πk) = sin (t)

Therefore, sin (π/3+ 2π) = sin (π/3)

From the unit circle, the value of sin (π/3)=√3/2

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