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Double-Angle and Half-Angle Formulas

In these lessons, we learn how to use the double angle formulas and the half-angle formulas to solve trigonometric equations and to prove trigonometric identities. We will also prove the double angle formulas and the half-angle formulas.


 
Related Topics:
More Lessons on Trigonometry
Trigonometry Worksheets
Trigonometry Games

The following figure gives the Double-Angle Formulas and Half-Angle Formulas. Scroll down the page for more examples and solutions of how to use, derive and proof the Double-Angle Formulas and Half-Angle Formulas.

Double-angle, Half-angle formulas

Example:

Given , find
a) sin 2θ
b)

Solution:

a) sin 2θ = 2 sin θ cos θ

Using Double Angle Identities to Solve Equations

Using Double Angle Identities to Solve Equations, Example 1.
This video uses some double angle identities for sine and/or cosine to solve some equations.
Example:
cos(4x) − 3cos(2x) = 4
Using Double Angle Identities to Solve Equations, Example 2
Example:
sin(2t) = sin(t)



Using Double Angle Identities to Solve Equations, Example 3
Example:
sin(2t) + 4sin(t) + 2cos(t) = −4

Prove Trigonometry Identities using Double Angles

Trigonometry Identities - Double Angles (1)
Example:
(1 − cos 2x)/sin 2x = tan x
Trigonometry Identities - Double Angle (2)
Example:
Prove tan x + cos x = 2 cosec 2x
Double-Angle Formulas
A few examples that use double-angle formulas from trigonometry.
Examples:
1. If cos x = 1/√10 with x in quadrant IV, find sin 2x
2. Graph y = 4 - 8 sin2x
3. Verify sin60° = 2sin30°cos30°


Proof of the Double Angle Formulae

Trigonometry - Proof of the Double Angle Formulae from the Addition Formulae.
How to derive the double angle identities for sine and cosine?
Examples:
If cosθ = -4/5 and sinθ > 0, find
a) sin2θ
b) cos2θ

Using Half-Angle Formulas

Half-Angle Formulas
A few problems involving half-angle formulas in trigonometry.
Examples:
Let sinA = 4/5 with A in Quadrant II.
Find
1. sin A/2
2. cos 2A
3. sec 2A
Half Angle Identities to Evaluate Trigonometric Expressions, Example 1
This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.
Example:
Find the exact value of sin(22.5°)

Half Angle Identities to Evaluate Trigonometric Expressions, Example 2
This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.
Example:
Find the exact value of tan(105°)
Half Angle Identities to Evaluate Trigonometric Expressions, Example 3
This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.
Find sin(a/2) if cos a = 3/5 for 0° ≤ a ≤ 90°

Proof Half-Angle Formulas

This video derives the half angle trigonometry identities for cosine, sine and tangent.
Proof of the Half Angle Formula


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