TIE - Applying Trigonometric Identities and Equations Overview

Applying Trigonometric Identities and Equations Overview

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!

In previous math courses, you have dealt with mostly right triangles. However, in this module we will explore non-right, or oblique, triangles! You can still use trigonometry to solve them, but not by your traditional trigonometric ratios. Oblique triangles are often used in navigation and estimating distances - read on to see how we will solve oblique triangles!

Essential Questions

What is an identity?
How do I use trigonometric identities to prove statements?
How do I use trigonometric identities to solve equations?
How can I calculate the area of any triangle given only two sides and a non-included angle?
How can I apply trigonometric relationships to non-right triangles?

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

Altitude of a triangle - The perpendicular distance between a vertex of a triangle and the side opposite that vertex.

Included Angle - The angle between two given sides of a triangle.

Law of Cosines - $LaTeX: c^2=a^2+b^2-2ab\cdot Cos(C)$ $c^2=a^2+b^2-2ab\cdot Cos(C)$ $c^{2} = a^{2} + b^{2} - 2 a b \cdot \cos C$

Law of Sines - $LaTeX: \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Oblique Triangle - A triangle that is not a right triangle.

Angle of Elevation - The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

Angle of Depression - The angle below horizontal that an observer must look to see an object that is lower than the observer.

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