Double-Angle and Half-Angle Identities: Easy Trigonometry Guide

 

Double-Angle and Half-Angle Identities

Unlock advanced trigonometry problems with confidence

Double-angle and half-angle identities help you rewrite difficult trigonometric expressions into forms that are easier to calculate. They are especially useful when an angle is doubled, divided by 2, or connected to a known sine, cosine, or tangent value.

The Big Idea

A double-angle identity changes an expression involving 2θ into expressions involving θ. A half-angle identity does the reverse: it finds a trigonometric value for θ ÷ 2 using a value for θ.

Double-Angle Identities

Sine Double-Angle Identity

sin(2θ) = 2 sin(θ) cos(θ)

Cosine Double-Angle Identities

cos(2θ) = cos²(θ) − sin²(θ)

cos(2θ) = 2cos²(θ) − 1

cos(2θ) = 1 − 2sin²(θ)

Tangent Double-Angle Identity

tan(2θ) = 2tan(θ) ÷ (1 − tan²(θ))

Worked Example 1: Find sin(2θ)

Suppose sin(θ) = 3/5 and cos(θ) = 4/5. Find sin(2θ).

Use sin(2θ) = 2sin(θ)cos(θ):

sin(2θ) = 2(3/5)(4/5) = 24/25

Worked Example 2: Find cos(2θ)

If sin(θ) = 3/5, use the cosine form that contains sine only.

Use cos(2θ) = 1 − 2sin²(θ):

cos(2θ) = 1 − 2(3/5)² = 1 − 18/25 = 7/25

Half-Angle Identities

Half-angle identities find the sine, cosine, or tangent of half an angle. The plus-or-minus sign is chosen according to the quadrant where the half-angle lies.

Sine and Cosine Half-Angle Identities

sin(θ ÷ 2) = ±√((1 − cos(θ)) ÷ 2)

cos(θ ÷ 2) = ±√((1 + cos(θ)) ÷ 2)

Tangent Half-Angle Identities

tan(θ ÷ 2) = ±√((1 − cos(θ)) ÷ (1 + cos(θ)))

tan(θ ÷ 2) = sin(θ) ÷ (1 + cos(θ))

Worked Example 3: Find a Half-Angle Value

Suppose θ is an acute angle and cos(θ) = 3/5. Find sin(θ ÷ 2).

Because θ is acute, θ ÷ 2 is also in a quadrant where sine is positive:

sin(θ ÷ 2) = √((1 − 3/5) ÷ 2)

= √((2/5) ÷ 2) = √(1/5) ≈ 0.447

How to Choose the Correct Identity

What you know	Best identity	Why
sin(θ) and cos(θ)	sin(2θ)	The formula uses both values directly.
Only sin(θ)	cos(2θ) = 1 − 2sin²(θ)	No cosine value is needed.
Only cos(θ)	cos(2θ) = 2cos²(θ) − 1	No sine value is needed.
cos(θ) and a half-angle	Cosine half-angle	Use the matching plus sign when appropriate.

On a phone, swipe horizontally inside the table if needed.

Common Mistakes to Avoid

Forgetting the angle change: Double-angle formulas use 2θ, while half-angle formulas use θ ÷ 2.

Choosing the wrong sign: Decide whether the half-angle is in a quadrant where the value is positive or negative.

Squaring the wrong quantity: In 1 − 2sin²(θ), square sin(θ) before multiplying by 2.

Calculator mode error: Use degree mode for degree-based problems.

Practice Problems

1. If sin(θ) = 5/13 and cos(θ) = 12/13, find sin(2θ).

2. If cos(θ) = 4/5, find cos(2θ) using 2cos²(θ) − 1.

3. If θ is acute and cos(θ) = 1/2, find cos(θ ÷ 2).

4. Which identity is best for finding cos(2θ) when sin(θ) is known?

Tap to reveal the answers

1. sin(2θ) = 2(5/13)(12/13) = 120/169. 2. cos(2θ) = 2(4/5)² − 1 = 32/25 − 1 = 7/25. 3. cos(θ ÷ 2) = √((1 + 1/2) ÷ 2) = √(3/4) = √3 ÷ 2. 4. cos(2θ) = 1 − 2sin²(θ).

Final Takeaway

Double-angle identities simplify expressions involving 2θ. Half-angle identities simplify expressions involving θ ÷ 2. Match the formula to the information you know, choose the correct sign from the quadrant, and check your calculator mode.