If cos x sin20 x and 0 x 90 the value of x is

To determine the value of x satisfying the equation cos(x) = sin(20x) within the range 0 < x < 90, we can start by using the identity relating cosine and sine.

Recall that sin(θ) = cos(90° – θ). Thus, we can rewrite the equation as:

cos(x) = cos(90° – 20x)

This leads us to two potential cases:

1. x = 90° – 20x
2. x = 180° – (90° – 20x)

Let’s solve the first case:

x = 90° – 20x
21x = 90°
x = 90°/21 = 4.2857°

Next, let’s solve the second case:

x = 180° – (90° – 20x)
x = 90° + 20x
-19x = 90°
x = -90°/19, which is not within the required range.

Thus, the only valid solution for 0 < x < 90 is:

x ≈ 4.29°

This value satisfies the original equation in the specified interval.