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Double-Angle and Half-Angle Identities

A free Trigonometry lesson from the “Trig Identities” unit, with a worked example and practice problems including step-by-step solutions.

Double-angle identities rewrite functions of 2x using functions of x: sin 2x = 2 sin x cos x, cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x, and tan 2x = 2 tan x / (1 - tan^2 x). Half-angle identities go the other way, giving sin(x/2) = +/- sqrt((1 - cos x)/2) and cos(x/2) = +/- sqrt((1 + cos x)/2), where the sign is chosen by the quadrant of x/2. Together they let you find exact values such as sin 15 (half of 30) and compute sin 2x or cos 2x from one ratio plus a quadrant.

What you'll learn

Why it matters: Double-angle identities collapse the doubled frequencies that appear when sound waves or AC signals multiply together, and half-angle formulas give engineers exact values for angles between the standard ones used in surveying and optics.

Worked example

Problem. Given sin x = 3/5 with x in Quadrant I, find sin 2x and cos 2x.

  1. Find cos x: since x is in QI, cos x = sqrt(1 - 9/25) = 4/5.
  2. sin 2x = 2 sin x cos x = 2(3/5)(4/5) = 24/25.
  3. cos 2x = 1 - 2 sin^2 x = 1 - 2(9/25) = 7/25.

Answer: sin 2x = 24/25, cos 2x = 7/25

Practice problems

1. Write the double-angle formula for sin 2x in terms of sin x and cos x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Write the double-angle formula for sin 2x in terms of sin x and cos x.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The double-angle identity for sine is sin 2x = 2 sin x cos x.
  4. It comes from the sum formula sin(x + x).
  5. Check the result by substituting or estimating: the response should match 2 sin x cos x and make sense in the original problem.

Answer: 2 sin x cos x

2. Which expression is NOT equal to cos 2x?

Choices: cos^2 x - sin^2 x · 2 cos^2 x - 1 · 1 - 2 sin^2 x · 2 sin x cos x

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which expression is NOT equal to cos 2x?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The three valid forms of cos 2x are cos^2 x - sin^2 x, 2cos^2 x - 1, and 1 - 2sin^2 x.
  4. 2 sin x cos x is the formula for sin 2x, not cos 2x.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2 sin x cos x

3. If sin x = 3/5 and x is in Quadrant I, find sin 2x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: If sin x = 3/5 and x is in Quadrant I, find sin 2x.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. cos x = sqrt(1 - 9/25) = 4/5 (positive in QI).
  4. sin 2x = 2(3/5)(4/5) = 24/25.
  5. Check the result by substituting or estimating: the response should match 24/25 and make sense in the original problem.

Answer: 24/25

4. If sin x = 3/5 and x is in Quadrant I, find cos 2x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: If sin x = 3/5 and x is in Quadrant I, find cos 2x.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Use cos 2x = 1 - 2 sin^2 x = 1 - 2(9/25).
  4. = 1 - 18/25 = 7/25.
  5. Check the result by substituting or estimating: the response should match 7/25 and make sense in the original problem.

Answer: 7/25

5. If cos x = 1/3, find cos 2x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: If cos x = 1/3, find cos 2x.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Use cos 2x = 2 cos^2 x - 1 (only cos x is needed).
  4. = 2(1/9) - 1 = 2/9 - 1 = -7/9.
  5. Check the result by substituting or estimating: the response should match -7/9 and make sense in the original problem.

Answer: -7/9

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