Even and Odd Trigonometric Identities

Learn how cosine and secant are even functions while sine, tangent, cosecant, and cotangent are odd functions.

Advanced25 minLesson
Printable worksheet

Definition

Trigonometric functions can be classified as even or odd based on how they behave when we negate the input angle.
Even Functions (symmetric about the -axis):
Odd Functions (symmetric about the origin):
These identities are called even-odd identities because they mirror the algebraic definitions of even and odd functions:
  • Even:
  • Odd:

Try it now

Which trigonometric function is even?

Worked Examples

Simplify:

1

Identify the function type

Cosine is an even function →

2

Apply the even identity

→ Angle becomes positive

3

Evaluate the cosine

→

Common Mistakes

Thinking all trig functions are odd

Why it's wrong: Students sometimes assume the negative sign always moves outside, forgetting that cosine and secant are even.

Correct: Remember: Cosine and secant are EVEN (negative disappears), while sine, tangent, cosecant, and cotangent are ODD (negative moves outside).

Writing

Why it's wrong: Confusing the even identity with odd functions. Cosine is even, not odd!

Correct: (no negative sign). The graph of cosine is symmetric about the -axis.

Forgetting to apply both identities in expressions

Why it's wrong: When simplifying , students may only transform one function.

Correct: Apply identities to ALL functions:

Confusing with

Why it's wrong: These are equal by the odd identity, but students may not recognize this equivalence.

Correct: . The negative can be inside or outside - they are equivalent for odd functions.

Interactive Visual

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sin(θ)

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cos(θ)

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tan(θ)

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Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Linear Function Explorer

y = x
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Y-Intercept (b)0
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Practice Problems

16 problems
Problem 1 of 16
Easy

Which trigonometric function is even?

Why It Matters

Even-odd identities are essential tools for:
  • Simplifying expressions: Replace with to work with positive angles
  • Solving equations: Transform equations with negative angles into standard form
  • Integration: Determine when integrals over symmetric intervals equal zero
  • Graphing: Understand the symmetry of trigonometric graphs
  • Physics: Analyze periodic motion, waves, and oscillations
These identities also build intuition for function behavior that extends to calculus and beyond.

Real World Applications

Signal Processing and Waves

In electronics and acoustics, understanding even-odd symmetry helps analyze and filter signals efficiently.

Example:

A sound wave can be decomposed into even (cosine) and odd (sine) components. If a signal is purely even, like , then , meaning it looks the same forwards and backwards.

1Try It Yourself

An audio engineer knows that for any frequency .

If a signal is , what is ?

Step 1: Write the mathematical expression

Apply the even identity:

Physics: Symmetric Forces

Many physical quantities depend on whether forces or fields are even or odd functions of position.

Example:

The gravitational force on a pendulum depends on . Since sine is odd, the force reverses direction when the pendulum swings to the opposite side: .

2Try It Yourself

A spring force is modeled by where is displacement.

Show that , confirming the force is restorative.

Step 1: Write the mathematical expression

Find :

Computer Graphics and Animation

Even-odd properties help optimize calculations for symmetric animations and reflections.

Example:

When rendering a symmetric shape, knowing that means you only need to calculate half the rotation angles.

3Try It Yourself

An animation rotates an object using and .

What are the coordinates at angle in terms of and ?

Step 1: Write the mathematical expression

Find :

Key Takeaways

  • 1Even functions satisfy : cosine and secant
  • 2Odd functions satisfy : sine, tangent, cosecant, and cotangent
  • 3Even functions have symmetry about the -axis
  • 4Odd functions have symmetry about the origin (180-degree rotational symmetry)
  • 5These identities help simplify expressions with negative angles
  • 6Memory aid: Only functions starting with 'co' that are even are cosine and secant

Frequently Asked Questions

Remember: Cosine and Secant are the only EVEN functions (both have 'c' and relate to the -coordinate). Everything else (Sine, Tangent, Cosecant, Cotangent) is ODD. Alternatively, graph them - even functions are symmetric about the -axis.
Remember: Cosine and Secant are the only EVEN functions (both have 'c' and relate to the -coordinate). Everything else (Sine, Tangent, Cosecant, Cotangent) is ODD. Alternatively, graph them - even functions are symmetric about the -axis.
On the unit circle, gives the -coordinate and gives the -coordinate. When you negate the angle (go clockwise instead of counterclockwise), the -coordinate stays the same but the -coordinate flips sign.
Since and , both inherit the odd property from sine in their definitions. The even cosine in cotangent's numerator is divided by odd sine, making the result odd.

Glossary

Even function
A function where for all ; symmetric about the -axis
Odd function
A function where for all ; symmetric about the origin
Identity
An equation that is true for all values of the variable
Unit circle
A circle with radius 1 centered at the origin, used to define trigonometric functions

Formula Card

Cosine (Even)

Cosine of a negative angle equals cosine of the positive angle

Secant (Even)

Secant of a negative angle equals secant of the positive angle

Sine (Odd)

Sine of a negative angle equals the negative of sine of the positive angle

Tangent (Odd)

Tangent of a negative angle equals the negative of tangent of the positive angle

Cosecant (Odd)

Cosecant of a negative angle equals the negative of cosecant of the positive angle

Cotangent (Odd)

Cotangent of a negative angle equals the negative of cotangent of the positive angle

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