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Trigonometric Identities

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Trigonometric Identities

Study Guide

Key Definition

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved, such as $\sin^2(x) + \cos^2(x) = 1$.

Important Notes

The Pythagorean identity is $\sin^2(x) + \cos^2(x) = 1$
Reciprocal identities include $\sin(x) = \frac{1}{\csc(x)}, \cos(x) = \frac{1}{\sec(x)}, \tan(x) = \frac{1}{\cot(x)}$
Co-function identities involve angles like $\sin(\frac{\pi}{2} - x) = \cos(x)$
Even-odd identities include $\sin(-x) = -\sin(x)$
Sum and difference formulas are important for calculating exact values
Other identities such as double-angle, half-angle, sum-to-product, and product-to-sum are beyond this scope but can be explored separately

Mathematical Notation

$\sin(x)$ is the sine function

$\cos(x)$ is the cosine function

$\tan(x)$ is the tangent function

$\csc(x)$ is the cosecant function

$\sec(x)$ is the secant function

$\cot(x)$ is the cotangent function

Remember to use proper notation when solving problems

Why It Works

Trigonometric identities simplify and solve complex trigonometric equations using known relationships like $\sin^2(x) + \cos^2(x) = 1$.

Remember

Always verify identities using fundamental formulas like $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

Quick Reference

Pythagorean Identity:$\sin^2(x) + \cos^2(x) = 1$

Reciprocal Identity:$\sin(x) = \frac{1}{\csc(x)}$

Co-function Identity:$\sin(\frac{\pi}{2} - x) = \cos(x)$

Understanding Trigonometric Identities

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Video explanation of this concept

Beginner

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Beginner Explanation

The Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ comes from considering a right triangle inscribed in the unit circle. Since any point on the circle satisfies x² + y² = 1, taking x = cos(x) and y = sin(x) gives the identity. This simple relationship forms the foundation of many trigonometric proofs.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which identity is correct? $\sin^2(x) + \cos^2(x)$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder uses an incline with a height of 3 ft and a hypotenuse of 5 ft. If the ramp is 3 ft high and the hypotenuse is 5 ft, find $\theta$ using the sine relationship.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove $\tan(x) = \frac{\sin(x)}{\cos(x)}$ using identities.

Challenge Quiz

Single Choice Quiz

Advanced

Find the value of $\cos(\frac{\pi}{2} - x)$.

Recap

Watch & Learn

Review key concepts and takeaways