Examples With Trigonometric Functions: Even, Odd Or Neither

In trigonometry, even and odd functions refer to the symmetry properties of trigonometric functions.

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:

A function f(x) is even if f(-x) = f(x) for all x in its domain.
An even function is symmetric (by reflection) about the y-axis.

1. Cosine (cos):
   cos(-x) = cos(x)
   Therefore, cosine is an even function.
   Graphically, the cosine function is symmetric with respect to the y-axis.
   
2. Secant (sec):
   sec(-x) = sec(x)
   Since secant is 1/cosine, it also inherits the even properties of cosine.
   Therefore, secant is an even function.
   

Odd Trigonometric Functions:

A function f(x) is odd if f(-x) = -f(x) for all x in its domain.
An odd function is symmetric (by 180° rotation) about the origin, i.e.

1. Sine (sin):
   sin(-x) = -sin(x)
   Therefore, sine is an odd function.
   Graphically, the sine function is symmetric with respect to the origin.
   
2. Tangent (tan):
   tan(-x) = -tan(x)
   Therefore, tangent is an odd function.
   Since tangent is sine/cosine, and sine is odd and cosine is even, tangent will be odd.
   
3. Cosecant (csc):
   csc(-x) = -csc(x)
   Since cosecant is 1/sine, it also inherits the odd properties of sine.
   Therefore, cosecant is an odd function.
   
4. Cotangent (cot):
   cot(-x) = -cot(x)
   Since cotangent is 1/tangent, it also inherits the odd properties of tangent.
   Therefore, cotangent is an odd function.
   

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

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Examples With Trigonometric Functions: Even, Odd Or Neither

Example 2

Determine whether the following trigonometric function is Even, Odd or Neither
a) f(x) = sec x tan x

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Example 3

b) g(x) = x⁴ sin x cos²x

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Example 4

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

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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)

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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) = ___.

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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)

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