Question

Use a double-angle identity to solve cos(2x) + 2sin2 x − sin x = 0 in [0,360°)

Answer 1

To solve the equation using a double-angle identity, replace cos(2x) with 1 - 2sin^2x. This simplifies to sinx = 1, which has the solution x = 90 degrees within the interval [0,360 degrees).

Step-by-step explanation:

To solve the equation cos(2x) + 2sin2x − sinx = 0 using a double-angle identity, we can replace cos(2x) with one of its equivalent identities. The double-angle identities for cosine are:

• cos(2x) = cos2x - sin2x
• cos(2x) = 2cos2x - 1
• cos(2x) = 1 - 2sin2x

Since the equation already contains sin2x, we can use the third identity:

1 - 2sin2x + 2sin2x - sinx = 0

This simplifies to:

1 - sinx = 0

sinx = 1

Now we solve for x within the interval [0,360°). The sine function is equal to 1 at 90°, so:

x = 90°