How do you solve 2 cos squared x plus sin x-1?

Answer 1

1

Answer 2

To solve 2 cos2x + sin x - 1 make use of the identity cos2x + sin2x = 1, then:

cos2x = 1 - sin2x

and

2 cos2x + sin x - 1 = 2 (1 - sin2x) + sin x - 1

= 2 - 2 sin2x + sin x - 1

= 1 + sin x - 2 sin2 x

if you let y = sin x, then you can see you have a quadratic:

1 + sin x - 2 sin2 x = 1 + y - 2y2

which can be solved for 1 + y - 2y2 = 0:

1 + y - 2y2 = 0

⇒ 2y2 - y - 1 = 0

⇒ (2y + 1)(y - 1) = 0

⇒ y = -1/2 or 1

but y = sin x, so:

sin x = -1/2 or sin x = 1

and so solve these for x.