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In this lesson we will look at how to determine whether a Trigonometric Function is Even, Odd or Neither.

**What is an even function?**

1. An even function is symmetric about the y-axis.

2. f(-x) = f(x)

**What is an odd function?**

1. An odd function is symmetric about the origin.

2. f(-x) = -f(x)

The following table shows the Even Trigonometric Functions and Odd Trigonometric Functions. Scroll down the page for more examples and step by step solutions.

**Even Trigonometric Functions and Identities**

Cosine function is even. cos(-x) = cos x

Secant function is even. sec(-x) = sec x

**Odd Trigonometric Functions and Identities**

Sine function is odd. sin(-x) = - sin x

Cosecant function is odd. csc(-x) = - csc x

Tangent function is odd. tan(-x) = - tan x

Cotangent function is odd. cot(-x) = - cot x

**Determine whether a trigonometric function is odd, even, or neither**

Examples with Trigonometric Functions: Even, Odd or Neither

Cosine function, Secant function, Sine function, Cosecant function, Tangent function, and Cotangent function

**Example 2**

Determine whether the following trigonometric function is Even, Odd or Neither

a) f(x) = sec x tan x Examples with Trigonometric Functions: Even, Odd or Neither

Example 3

b) g(x) = x^{4} sin x cos^{2}x
Examples with Trigonometric Functions: Even, Odd or Neither

Example 4

c) h(x) = cos x + sin x**How to use the even-odd properties of the trigonometric functions?**

Example: Find the exact value using even-odd properties.

(a) sin(-30°)

(b) cos(-3π/4)

(c) tan(-π/4)**How to determine trig function values based upon whether the function is odd or even?**

Determine each function value.

If cos(x) = 0.5, then cos(-x) = ___.

If sin(x) = 0.15, then sin(-x) = ___.

If tan(-x) = -3, then tan(x) = ___.

If sec(-x) = 1.4, then sec(x) = ___.**How to use even or odd properties to evaluate trig functions?**

Evaluate the trigonometric function by first using even/odd properties to rewrite the expression with a positive angle. Give an exact answer Do not use a calculator.

sin(-45°)

sec(210°)

cos(-π6)

csc(-3π/2)

**Trigonometry Calculator with step by step solutions**

Right Triangle Trigonometry, Radian Measure and Circular Functions

Graphing Trigonometric Functions, Simplifying Trigonometric Expressions

Verifying Trigonometric Identities, Verifying Trigonometric Identities

Using Fundamental Identities, Solving Trigonometric Equations

Complex Numbers, Analytic Geometry in Polar Coordinates

Exponential and Logarithmic Functions, Vector Arithmetic, Vectors

More Lessons for Trigonometry

Math Worksheets

In this lesson we will look at how to determine whether a Trigonometric Function is Even, Odd or Neither.

1. An even function is symmetric about the y-axis.

2. f(-x) = f(x)

1. An odd function is symmetric about the origin.

2. f(-x) = -f(x)

The following table shows the Even Trigonometric Functions and Odd Trigonometric Functions. Scroll down the page for more examples and step by step solutions.

Cosine function is even. cos(-x) = cos x

Secant function is even. sec(-x) = sec x

Sine function is odd. sin(-x) = - sin x

Cosecant function is odd. csc(-x) = - csc x

Tangent function is odd. tan(-x) = - tan x

Cotangent function is odd. cot(-x) = - cot x

Examples with Trigonometric Functions: Even, Odd or Neither

Cosine function, Secant function, Sine function, Cosecant function, Tangent function, and Cotangent function

Determine whether the following trigonometric function is Even, Odd or Neither

a) f(x) = sec x tan x Examples with Trigonometric Functions: Even, Odd or Neither

Example 3

b) g(x) = x

Example 4

c) h(x) = cos x + sin x

Example: Find the exact value using even-odd properties.

(a) sin(-30°)

(b) cos(-3π/4)

(c) tan(-π/4)

Determine each function value.

If cos(x) = 0.5, then cos(-x) = ___.

If sin(x) = 0.15, then sin(-x) = ___.

If tan(-x) = -3, then tan(x) = ___.

If sec(-x) = 1.4, then sec(x) = ___.

Evaluate the trigonometric function by first using even/odd properties to rewrite the expression with a positive angle. Give an exact answer Do not use a calculator.

sin(-45°)

sec(210°)

cos(-π6)

csc(-3π/2)

Right Triangle Trigonometry, Radian Measure and Circular Functions

Graphing Trigonometric Functions, Simplifying Trigonometric Expressions

Verifying Trigonometric Identities, Verifying Trigonometric Identities

Using Fundamental Identities, Solving Trigonometric Equations

Complex Numbers, Analytic Geometry in Polar Coordinates

Exponential and Logarithmic Functions, Vector Arithmetic, Vectors

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