Question

Express sin5x in powers of sin x

Answer 1

Show that Sin 5 theta=5Sin theta -20 Sin^3 theta +16 Sin^5 theta?

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Adrija Dalal, B. Sc Vivekananda College, Thakurpukur, Aligunj Rishi Rajnarayana Balika Vidyalaya (2020)

Answered Feb 21, 2018

Here theta is denoted by A

sin5A = sin(3A+2A)

= sin3Acos2A+cos3Asin2A

=(3sinA-4sin³A)(1–2sin²A) + cos(2A+A)sin2A

=3sinA-10sin³A+8sin^5A+ [cos2AcosA- sin2AsinA]sin2A

=3sinA-10sin³A+8sin^5A+ [(1–2sin²A)cosA- 2sin²AcosA]2sinAcosA

=3sinA-10sin³A+8sin^5A+[cosA-4sin²AcosA]2sinAcosA

=3sinA-10sin³A+8sin^5A+2sinAcos²A- 8sin³Acos²A

=3sinA-10sin³A+8sin^5A+2sinA(1-sin²A)-

8sin³A(1-sin²A)

=3sinA-10sin³A+8sin^5A+2sinA-2sin³A-8sin³A+8sin^5A

=5sinA-20sin³A+16sin^5A[Proved]