Even and Odd Trigonometric Identities

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

Advanced25 minLesson
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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 yy-axis):
cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta)
sec⁡(−θ)=sec⁡(θ)\sec(-\theta) = \sec(\theta)
Odd Functions (symmetric about the origin):
sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)
tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta)
csc⁡(−θ)=−csc⁡(θ)\csc(-\theta) = -\csc(\theta)
cot⁡(−θ)=−cot⁡(θ)\cot(-\theta) = -\cot(\theta)
These identities are called even-odd identities because they mirror the algebraic definitions of even and odd functions:
  • Even: f(−x)=f(x)f(-x) = f(x)
  • Odd: f(−x)=−f(x)f(-x) = -f(x)

Try it now

Which trigonometric function is even?

Worked Examples

Simplify: cos⁡(−60°)\cos(-60°)

1

Identify the function type

Cosine is an even function → cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta)

2

Apply the even identity

cos⁡(−60°)=cos⁡(60°)\cos(-60°) = \cos(60°) → Angle becomes positive

3

Evaluate the cosine

cos⁡(60°)=12\cos(60°) = \frac{1}{2}

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 cos⁡(−x)=−cos⁡(x)\cos(-x) = -\cos(x)

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

Correct: cos⁡(−x)=cos⁡(x)\cos(-x) = \cos(x) (no negative sign). The graph of cosine is symmetric about the yy-axis.

Forgetting to apply both identities in expressions

Why it's wrong: When simplifying sin⁡(−x)cos⁡(−x)\frac{\sin(-x)}{\cos(-x)}, students may only transform one function.

Correct: Apply identities to ALL functions: sin⁡(−x)cos⁡(−x)=−sin⁡(x)cos⁡(x)=−tan⁡(x)\frac{\sin(-x)}{\cos(-x)} = \frac{-\sin(x)}{\cos(x)} = -\tan(x)

Confusing −sin⁡(x)-\sin(x) with sin⁡(−x)\sin(-x)

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

Correct: sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x). The negative can be inside or outside - they are equivalent for odd functions.

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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 sin⁡(−θ)\sin(-\theta) with −sin⁡(θ)-\sin(\theta) 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 f(t)=cos⁡(2πft)f(t) = \cos(2\pi ft), then f(−t)=f(t)f(-t) = f(t), meaning it looks the same forwards and backwards.

1Try It Yourself

An audio engineer knows that cos⁡(−ωt)=cos⁡(ωt)\cos(-\omega t) = \cos(\omega t) for any frequency ω\omega.

If a signal is S(t)=cos⁡(440t)S(t) = \cos(440t), what is S(−t)S(-t)?

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 sin⁡(θ)\sin(\theta). Since sine is odd, the force reverses direction when the pendulum swings to the opposite side: F(−θ)=−F(θ)F(-\theta) = -F(\theta).

2Try It Yourself

A spring force is modeled by F(x)=−ksin⁡(x)F(x) = -k\sin(x) where xx is displacement.

Show that F(−x)=−F(x)F(-x) = -F(x), confirming the force is restorative.

Step 1: Write the mathematical expression

Find F(−x)F(-x):

Computer Graphics and Animation

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

Example:

When rendering a symmetric shape, knowing that cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta) means you only need to calculate half the rotation angles.

3Try It Yourself

An animation rotates an object using x(θ)=cos⁡(θ)x(\theta) = \cos(\theta) and y(θ)=sin⁡(θ)y(\theta) = \sin(\theta).

What are the coordinates at angle −θ-\theta in terms of cos⁡(θ)\cos(\theta) and sin⁡(θ)\sin(\theta)?

Step 1: Write the mathematical expression

Find (x(−θ),y(−θ))(x(-\theta), y(-\theta)):

Key Takeaways

  • 1Even functions satisfy f(−x)=f(x)f(-x) = f(x): cosine and secant
  • 2Odd functions satisfy f(−x)=−f(x)f(-x) = -f(x): sine, tangent, cosecant, and cotangent
  • 3Even functions have symmetry about the yy-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 xx-coordinate). Everything else (Sine, Tangent, Cosecant, Cotangent) is ODD. Alternatively, graph them - even functions are symmetric about the yy-axis.
Remember: Cosine and Secant are the only EVEN functions (both have 'c' and relate to the xx-coordinate). Everything else (Sine, Tangent, Cosecant, Cotangent) is ODD. Alternatively, graph them - even functions are symmetric about the yy-axis.
On the unit circle, cos⁡(θ)\cos(\theta) gives the xx-coordinate and sin⁡(θ)\sin(\theta) gives the yy-coordinate. When you negate the angle (go clockwise instead of counterclockwise), the xx-coordinate stays the same but the yy-coordinate flips sign.
Since cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} and csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}, 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 f(−x)=f(x)f(-x) = f(x) for all xx; symmetric about the yy-axis
Odd function
A function where f(−x)=−f(x)f(-x) = -f(x) for all xx; 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)

cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta)

Cosine of a negative angle equals cosine of the positive angle

Secant (Even)

sec⁡(−θ)=sec⁡(θ)\sec(-\theta) = \sec(\theta)

Secant of a negative angle equals secant of the positive angle

Sine (Odd)

sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)

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

Tangent (Odd)

tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta)

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

Cosecant (Odd)

csc⁡(−θ)=−csc⁡(θ)\csc(-\theta) = -\csc(\theta)

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

Cotangent (Odd)

cot⁡(−θ)=−cot⁡(θ)\cot(-\theta) = -\cot(\theta)

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

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