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Basic Trigonometric Identities

Master basic trigonometric identities with interactive lessons and practice problems! Designed for high-school students preparing for upper-level work.

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Basic Trigonometric Identities

Study Guide

Key Definition

The basic trigonometric identities are equations involving the trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) that are always true whenever the expressions involved are defined.

Important Notes

These identities can be derived from the Pythagorean Theorem or from the unit-circle definition of the trigonometric functions.
The reciprocal identities give equivalent equations for the trigonometric functions.
Remember that $ \tan(x), \sec(x) $ are undefined when $ \cos(x)=0 $ and $ \cot(x), \csc(x) $ are undefined when $ \sin(x)=0.$

Mathematical Notation

$\sin^2(x) + \cos^2(x) = 1$ is one of the most commonly used trigonometric identities.

$1 + \tan^2(x) = \sec^2(x)$ and $1 + \cot^2(x) = \csc^2(x)$ are also key identities.

Use proper notation when writing squared trigonometric expressions; $\sin^2(x)$ means $(\sin x)^2$, not $\sin(x^2).$

Why It Works

These identities come from the Pythagorean Theorem applied to the unit circle: for any point $(\cos x,\, \sin x)$ on the unit circle, $\cos^2 x + \sin^2 x = 1.$ Dividing by $\cos^2 x$ or $\sin^2 x$ produces the other identities.

Remember

Simplify trigonometric expressions with these identities before attempting algebraic manipulation or calculator evaluation.

Quick Reference

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

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

Understanding Basic Trigonometric Identities

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

These identities are fundamental to trigonometry. They let you replace one trigonometric expression with another that may be easier to work with.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the value of $\sin^2(x) + \cos^2(x)$?

Real-World Problem

Question Exercise

Intermediate

Physics Scenario

A pendulum's displacement from equilibrium is modeled by $\theta(t) = \sin(t)$ while its velocity is modeled by $\phi(t) = \cos(t)$. Evaluate $\theta^2(t) + \phi^2(t)$ for any time $t$.

Advanced Challenge

Thinking Exercise

Advanced

Think About This

Prove that $1 + \tan^2(x) = \sec^2(x)$.

Recap

Watch & Learn

Review key concepts and takeaways