The Cofunction and Odd-Even Identities

Study Guide

Key Definition

The cofunction identities relate trigonometric functions for complementary angles: $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$, $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$, $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$, $\cot\left(\frac{\pi}{2} - x\right) = \tan(x)$, $\sec\left(\frac{\pi}{2} - x\right) = \csc(x)$, and $\csc\left(\frac{\pi}{2} - x\right) = \sec(x)$.

Important Notes

$\sin(-x) = -\sin(x)$ shows sine is an odd function
$\cos(-x) = \cos(x)$ shows cosine is an even function
$\tan(-x) = -\tan(x)$
$\cot(-x) = -\cot(x)$
$\csc(-x) = -\csc(x)$
$\sec(-x) = \sec(x)$
Cofunction identities can simplify calculations
Remember symmetry in trigonometric functions

Mathematical Notation

$\sin(\theta)$ is the sine function

$\cos(\theta)$ is the cosine function

$\tan(\theta)$ is the tangent function

$\cot(\theta)$ is the cotangent function

$\sec(\theta)$ is the secant function

$\csc(\theta)$ is the cosecant function

Remember to use proper notation when solving problems

Why It Works

Cofunction identities are based on the complementary angles in right triangles and the unit circle, where $\frac{\pi}{2}$ radians or $90^\circ$ represents a right angle.

Remember

The key formulas for cofunction and odd-even identities help in simplifying trigonometric expressions: $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$ and $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$.

Quick Reference

Cofunction Identity:$\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$

Cofunction Identity:$\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$

Cofunction Identity:$\sec\left(\frac{\pi}{2} - x\right) = \csc(x)$

Odd Function:$\sin(-x) = -\sin(x), \tan(-x) = -\tan(x), \cot(-x) = -\cot(x), \csc(-x) = -\csc(x)$

Even Function:$\cos(-x) = \cos(x), \sec(-x) = \sec(x)$

Understanding The Cofunction and Odd-Even Identities

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When two angles are complementary (sum to $90^\circ$), their sine and cosine values swap: $\sin(\theta) = \cos(90^\circ - \theta)$ and $\cos(\theta) = \sin(90^\circ - \theta)$. For example, $\sin(30^\circ) = \cos(60^\circ) = 1/2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\sin(30^\circ)$ using the cofunction identity?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are building a ramp. If the angle of elevation is $30^\circ$, find the cofunction identity that relates to the $\sin$ of this angle.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\cos(\alpha) = \frac{1}{2}$, what is $\cos(-\alpha)$?

Challenge Quiz

Single Choice Quiz

Advanced

Given $\tan(45^\circ)$, find $\cot(45^\circ)$ using the cofunction identity.

Recap

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