All subjects · Further Mathematics · WFM02 Further Pure Mathematics 2

Powers of sin and cos as multiple angles

Edexcel IAL Further Mathematics · WFM02 Further Pure Mathematics 2

The reverse job, writing cosn θ or sinn θ in terms of cos kθ and sin kθ, uses z = eiθ. It is the key step for integrating powers of sin and cos.

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The two identities

With z = cos θ + i sin θ, de Moivre gives zn = cos nθ + i sin nθ and z-n = cos nθ - i sin nθ. So

zn + 1/zn = 2cos nθ and zn - 1/zn = 2i sin nθ.

In particular z + 1/z = 2cos θ and z - 1/z = 2i sin θ.

The method

1. Write (2cos θ)n = (z + 1/z)n, or (2i sin θ)n = (z - 1/z)n.

2. Expand with the binomial theorem.

3. Pair terms from the two ends: z4 + z-4 = 2cos 4θ, and so on.

4. Divide by 2n (or (2i)n) to finish.

Why it is useful

∫ cos4 θ dθ is hard directly, but cos4 θ = (1/8)(cos 4θ + 4cos 2θ + 3), and each term integrates at once. This is the main application in exams.

Worked example

Show that cos4 θ = (1/8)(cos 4θ + 4cos 2θ + 3).

  1. (2cos θ)4 = (z + 1/z)4 = z4 + 4z2 + 6 + 4z-2 + z-4.
  2. Pair the terms: (z4 + z-4) + 4(z2 + z-2) + 6 = 2cos 4θ + 8cos 2θ + 6.
  3. So 16cos4 θ = 2cos 4θ + 8cos 2θ + 6; divide by 16.

Answer: cos4 θ = (1/8)(cos 4θ + 4cos 2θ + 3)

Key idea: zn + z-n = 2cos nθ and zn - z-n = 2i sin nθ. Expand (z ± 1/z)n, pair the terms from the ends, then divide.

Check yourself

  1. Which expression equals sin3 θ?
    • (sin 3θ - 3sin θ)/4
    • (3sin θ + sin 3θ)/4
    • (3sin θ - sin 3θ)/8
    • (3sin θ - sin 3θ)/4
    Show answer

    (3sin θ - sin 3θ)/4. (2i sin θ)3 = (z - 1/z)3 = (z3 - z-3) - 3(z - z-1) = 2i sin 3θ - 6i sin θ. Since (2i)3 = -8i: -8i sin3 θ = 2i sin 3θ - 6i sin θ, so sin3 θ = (3sin θ - sin 3θ)/4.

  2. cos5 θ = a cos 5θ + b cos 3θ + c cos θ. Find b. Give your answer as a fraction in its simplest form.
    Show answer

    5/16. (z + 1/z)5 = (z5 + z-5) + 5(z3 + z-3) + 10(z + z-1) = 2cos 5θ + 10cos 3θ + 20cos θ. Divide by 32: b = 10/32 = 5/16.

  3. Use cos4 θ = (1/8)(cos 4θ + 4cos 2θ + 3) to find ∫ from 0 to π/2 of cos4 θ dθ, to 4 decimal places.
    Show answer

    0.5890. ∫ = (1/8)[sin 4θ/4 + 2sin 2θ + 3θ] from 0 to π/2 = (1/8)(3π/2) = 3π/16 = 0.5890.

  4. With z = cos θ + i sin θ, which expression equals zn - 1/zn?
    • 2i sin nθ
    • 2sin nθ
    • 2cos nθ
    • -2i sin nθ
    Show answer

    2i sin nθ. zn = cos nθ + i sin nθ and z-n = cos nθ - i sin nθ. Subtracting leaves 2i sin nθ.

  5. sin4 θ = a cos 4θ + b cos 2θ + c. Find b. Give your answer as a fraction in its simplest form.
    Show answer

    -1/2. (z - 1/z)4 = (z4 + z-4) - 4(z2 + z-2) + 6 = 2cos 4θ - 8cos 2θ + 6, and (2i)4 = 16. So sin4 θ = (1/8)(cos 4θ - 4cos 2θ + 3); the coefficient is -4/8 = -1/2.

  6. Using sin3 θ = (3sin θ - sin 3θ)/4, find the exact value of ∫ from 0 to π/2 of sin3 θ dθ. Give your answer as a fraction in its simplest form.
    Show answer

    2/3. (1/4)[-3cos θ + cos 3θ/3] from 0 to π/2 = (1/4)[0 - (-3 + 1/3)] = (1/4)(8/3) = 2/3.

  7. cos6 θ is written as a cos 6θ + b cos 4θ + c cos 2θ + d. Find the constant d. Give your answer as a fraction in its simplest form.
    Show answer

    5/16. (z + 1/z)6 has middle term C(6, 3) = 20. Dividing by 26 = 64 gives d = 20/64 = 5/16.

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Last updated 8 November 2026. Spotted a mistake? Tell us.