Find all solutions to the equation in the interval [0, 2π). cos x = sin 2x

Solution:

Given, cos x = sin 2x

We have to find the solutions to the equation in the interval (0, 2π).

We know, sin 2x = 2 sin x cos x

So, cos x = 2 sin x cos x

The equation can be rewritten as

2 sin x cos x - cos x = 0

cos x(2 sin x - 1) = 0

Now, cos x = 0

x = cos⁻¹(0)

x = π/2 and 3π/2

2 sin x - 1 = 0

2 sin x = 1

sin x = 1/2

x = sin⁻¹(1/2)

x = π/6 and 5π/6

Therefore, the values of x are π/2, 3π/2, π/6 and 5π/6.

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Find all solutions to the equation in the interval [0, 2π). cos x = sin 2x

Summary:

All solutions to the equation in the interval [0, 2π). cos x = sin 2x are π/2, 3π/2, π/6 and 5π/6.