Compound and double-angle formulae

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What this lesson covers

  • sin(A + B) = sin A cos B + cos A sin B.
  • cos(A + B) = cos A cos B − sin A sin B.
  • sin 2A = 2 sin A cos A, and cos 2A has three forms.
Key idea

The double angles

  • sin 2A = 2 sin A cos A.
  • cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A.
Worked example

An exact value of sin 75°

  • Split 75° into 45° + 30°.
  1. sin 45° cos 30° + cos 45° sin 30°
  2. = (1/√2)(√3/2) + (1/√2)(1/2)
  3. = (√6 + √2) ÷ 4
Worked example

Given sin A = 3/5, A acute

  • Use a 3, 4, 5 triangle to get cos A = 4/5.
  1. sin 2A = 2 × 3/5 × 4/5 = 24/25
  2. cos 2A = 1 − 2(9/25)
  3. cos 2A = 7/25

Common mistakes

  • Writing sin(A + B) = sin A + sin B.
    Sine does not split across a plus sign.
  • Using the wrong sign in cos(A + B).
    The cosine formula changes the sign.

Your turn

  • 1. Find cos 15° exactly.
  • 2. Given cos A = 5/13, A acute, find sin 2A.
  • 3. Find cos 2A.
Show answer

1. (√6 + √2) ÷ 4 2. 120/169 3. −119/169

Summary

What to take away

  • Split the angle into known pieces.
  • Choose the cos 2A form that suits the question.

More in AS 91575

Trig functions and radiansNotesTrig identities and proofsNotesCompound and double-angle formulaeNotesTrig equations and general solutionsNotesModelling periodic situationsNotes

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