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.

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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.

### 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 sin^{2}x

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**

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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.

**Example:**

Given , find

a) sin 2θ

b)

*Solution:*

a) sin 2θ = 2 sin θ cos θ

This video uses some double angle identities for sine and/or cosine to solve some equations.

Example:

cos(4x) − 3cos(2x) = 4

Example:

sin(2t) = sin(t)

Example:

sin(2t) + 4sin(t) + 2cos(t) = −4

Example:

(1 − cos 2x)/sin 2x = tan x

Example:

Prove tan x + cos x = 2 cosec 2x

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 sin

3. Verify sin60° = 2sin30°cos30°

Examples:

If cosθ = -4/5 and sinθ > 0, find

a) sin2θ

b) cos2θ

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

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°)

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°)

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°

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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