Trigonometry

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Let z be a complex number such that
z=2(cos8∘+icos82∘).

Then z^5 can be expressed as r(sinα∘+icosα∘), where r is a real number and 0≤α≤90. What is the value of r+α?

  • Trigonometry -

    since cos 82° = sin 8°,
    z^5 = 2^5 cis(5*8)
    = 32 cis 40°
    = 32 (cos40° + i sin40°)
    = 32 (sin50° + i cos50°)

    32+50 = 82

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