TI - Trigonometric Identities Module Overview

Trigonometric Identities Module Overview

Introduction

In this module, we will finish our exploration of trigonometry with a deep dive into trigonometric identities! We will solve more trigonometric equations and learn new identities. Make sure you keep your Unit Circle handy!

Essential Questions

What is an identity?
How do I use trigonometric identities to prove statements?
How do I use trigonometric identities to solve equations?

Trigonometric Identities Key Terms

The following key terms will help you understand the content in this module.

Addition Identity for Cosine - equation image indicator $LaTeX: \cos\left(x+y\right)=\cos x\cos y-\sin x\sin y$ $\cos\left(x+y\right)=\cos x\cos y-\sin x\sin y$

Addition Identity for Sine - equation image indicator $LaTeX: \sin\left(x+y\right)=\sin x\cos y+\cos x\sin y$ $\sin\left(x+y\right)=\sin x\cos y+\cos x\sin y$

Addition Identity for Tangent - equation image indicator $LaTeX: \tan\left(x+y\right)=\frac{\tan x+\tan y}{1-\tan x\tan y}$ $\tan\left(x+y\right)=\frac{\tan x+\tan y}{1-\tan x\tan y}$

Double Angle Identity for Sine - equation image indicator $LaTeX: \sin\left(2x\right)=2\sin x\cos x$ $\sin\left(2x\right)=2\sin x\cos x$

Double Angle Identity for Cosine - equation image indicator $LaTeX: \cos\left(2x\right)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$ $\cos\left(2x\right)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$

Double Angle Identity for Tangent - equation image indicator $LaTeX: \tan\left(2x\right)=\frac{2\tan x}{1-\tan^2x}$ $\tan\left(2x\right)=\frac{2\tan x}{1-\tan^2x}$

Half Angle Identity for Sine - equation image indicator $LaTeX: \sin\left(\frac{x}{2}\right)=\pm\frac{\sqrt[]{1-\cos x}}{2}$ $\sin\left(\frac{x}{2}\right)=\pm\frac{\sqrt[]{1-\cos x}}{2}$

Half Angle Identity for Cosine - equation image indicator $LaTeX: \cos\left(\frac{x}{2}\right)=\pm\frac{\sqrt[]{1+\cos x}}{2}$ $\cos\left(\frac{x}{2}\right)=\pm\frac{\sqrt[]{1+\cos x}}{2}$

Half Angle Identity for Tangent - equation image indicator $LaTeX: \tan\left(\frac{x}{2}\right)=\pm\sqrt[]{\frac{1-\cos x}{1+\cos x}}=\frac{\sin x}{1+\cos x}$ $\tan\left(\frac{x}{2}\right)=\pm\sqrt[]{\frac{1-\cos x}{1+\cos x}}=\frac{\sin x}{1+\cos x}$

Subtraction Identity for Cosine - equation image indicator $LaTeX: \cos\left(x-y\right)=\cos x\cos y+\sin x\sin y$ $\cos\left(x-y\right)=\cos x\cos y+\sin x\sin y$

Subtraction Identity for Sine - equation image indicator $LaTeX: \sin\left(x-y\right)=\sin x\cos y-\cos x\sin y$ $\sin\left(x-y\right)=\sin x\cos y-\cos x\sin y$

Subtraction Identity for Tangent - equation image indicator $LaTeX: \tan\left(x-y\right)=\frac{\tan x-\tan y}{1+\tan x\tan y}$ $\tan\left(x-y\right)=\frac{\tan x-\tan y}{1+\tan x\tan y}$

Even Function - a function with symmetry about the y-axis that satisfies the relationship equation image indicator $LaTeX: f\left(-x\right)=f\left(x\right)$ $f\left(-x\right)=f\left(x\right)$

Odd Function - a function with symmetry about the origin that satisfies the relationship equation image indicator $LaTeX: f\left(-x\right)=-f\left(x\right)$ $f\left(-x\right)=-f\left(x\right)$

Pythagorean Identities - $LaTeX: \cos^2\theta+\sin^2\theta=1\\ 1+\tan^2\theta=\sec^2\theta\\ 1+\cot^2\theta=\csc^2\theta\\$ $\cos^2\theta+\sin^2\theta=1\\ 1+\tan^2\theta=\sec^2\theta\\ 1+\cot^2\theta=\csc^2\theta\\$

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