Answer to Question #117676 in Complex Analysis for Kelvin addo

Question

Express sin 5 threat and cos5theta in the forms of sin theta

Answer 1

$sin5\theta$

$= sin(3\theta+2\theta)$

$= sin3\theta cos2\theta+cos3\theta sin2\theta$

$=(3sin\theta-4sin³\theta)(1–2sin²\theta) + cos(2\theta+\theta)sin2\theta$

$=3sin\theta-10sin³\theta+8sin^5\theta+ [cos2\theta cos\theta- sin2\theta sin\theta]sin2\theta$

$=3sin\theta-10sin³\theta+8sin^5\theta+ [(1–2sin²\theta)cos\theta- 2sin²\theta cos\theta]2sin\theta cos\theta$

$=3sin\theta-10sin³\theta+8sin^5\theta+[cos\theta-4sin²\theta cos\theta]2sin\theta cos\theta$

$=3sin\theta-10sin³\theta+8sin^5\theta+2sin\theta cos²\theta- 8sin³\theta cos²\theta$

$=3sin\theta-10sin³\theta+8sin^5\theta+2sin\theta(1-sin²\theta)-8sin³\theta(1-sin²\theta)$

$=3sin\theta-10sin³\theta+8sin^5\theta+2sin\theta-2sin³\theta-8sin³\theta+8sin^5\theta$

$=5sin\theta-20sin³\theta+16sin^5\theta$