cos(x) - 5 sin(2x) = 0

Question

Find all solutions on the interval
[0, 2π).
(Enter your answers as a comma-separated list. Round your answers to four decimal places. If an answer does not exist, enter DNE.)
cos(x) − 5 sin(2x) = 0

Answer 1

cos x - 5 sin 2x = 0

cos x - 5 (2 sin x cos x) = 0           double angle formula

cox x - 10 sin x cos x = 0

cos x (1 - 10 sin x) = 0 

 

Can you solve for x?

Answer 2

cos(x) − 5sin(2x) = 0

 

Use the identity sin(2x) = 2sin(x)cos(x)

 

cos(x) - 5[2sin(x)cos(x)] = 0

 

cos(x)[1 - 10sin(x)] = 0

 

Then cos(x) = 0 or 1 - 10sin(x) = 0

 

If cos(x) = 0, then x = π/2 or 3π/2  [You can convert to decimals.]

 

If 1 - 10sin(x) = 0, then

 

sin(x) = 1/10 ⇒ x =sin^-1(1/10) ≅ 0.10017 rad or π - 0.10017 rad  [You can convert to decimals.]