cos(2theta)=sin(theta)

Question

solve the following 0<(or equal to) theta<2pi

Answer 1

cos (2x) has a lot of equivalent ways it can be written, but a convenient way for us would involve only sines since that makes the equation a lot easier to work with.

One of the identities for cos 2x is cos 2x = 1 - 2 sin²(x).

This means an equivalent way of writing our problem is: 1 - 2sin²x = sin(x).

If we let y = sin(x), then the equation becomes a quadratic:

1 - 2y² = y , or more in a more familiar form, 1 - 2y² - y = 0 or -2y² - y + 1 = 0.

We have the quadratic formula to help us here:

y = -(-1) ± √((-1)² - 4(-2)(1)) / 2(-2)

y = 1 ± √(1 + 8) / -4

y = 1 ± √9 /-4

y = 1 ± 3 / - 4

y = 1+3/ -4 or y = 1 - 3 / -4

y = 4/-4 = -1 or y = -2/-4 = 1/2

Since y = sin x, we have two options here: either sin (x) = -1 or sin(x) = 1/2

sin(x) = -1 only has one solution on the given interval, x = 3π/2 radians or 270°

sin(x) = 1/2 has two solutions on this interval, namely x = π/6 radians and x = 5π/6 radians or x = 30° and x = 150°

This means overall, this problem has three solutions on the given interval: x = 3π/2 radians or 270°, x = π/6 radians or 30° and x = 5π/6 radians or 150°

Answer 2