What is the value of x if cos(x) = sin(20x) and 0 <= x <= 90 degrees?

by

To find the value of x in the equation cos(x) = sin(20x), we can start by utilizing the identity that relates sine and cosine: sin(A) = cos(90° – A). This allows us to rewrite the equation as follows:

 

cos(x) = cos(90° – 20x)

 

This implies two scenarios due to the properties of cosine:

  1. Scenario 1: x = 90° – 20x
  2. Scenario 2: x = 360° – (90° – 20x) (but this would exceed our specified range of 0° to 90°)

Now, let’s solve Scenario 1:

x + 20x = 90°

21x = 90°

x = 90° / 21

x = 4.2857° (approximately)

 

Now, it is important to check if this solution satisfies the original equation:

We find that:

cos(4.2857°) is approximately equal to 0.9962

sin(20 * 4.2857°) is approximately equal to 0.9962

Since both values are approximately equal, we confirm that the solution is correct.

 

Thus, the value of x that satisfies the equation cos(x) = sin(20x) in the range 0° ≤ x ≤ 90° is approximately 4.29°.

Leave a Comment