Solving the Trigonometric Equation cos2x sin x 30°

This article will guide you through the process of solving the trigonometric equation cos2x sin x 30°. We will use trigonometric identities and algebraic manipulation to break down the problem step by step. The solution will include checking specific angles and, if necessary, employing numerical or graphical methods to find approximate solutions.

Step-by-Step Solution

Step 1: Use the Double Angle Identity for cosine

Recall the double angle identity for cosine:

cos2x 1 - 2sin^2x

By substituting this identity into the equation, we get:

1 - 2sin^2x sinx 30°

Step 2: Expand sin x 30° using the sine addition formula

The sine addition formula is:

sin(a b) sin a cos b cos a sin b

Given sin x 30°, we can expand it as:

sin x 30° sin x cos 30° cos x sin 30°

Since cos 30° √3/2 and sin 30° 1/2, we have:

sin x 30° sin x ( √3/2 ) cos x ( 1/2 )

Step 3: Substitute the expanded formula back into the equation

Substituting the expanded formula back into the equation gives:

1 - 2sin^2x sin x ( √3/2 ) cos x ( 1/2 )

Step 4: Rearrange the equation

Rearranging the equation yields:

2sin^2x - sin x ( √3/2 ) - cos x ( 1/2 ) - 1 0

Step 5: Consider solving for specific angles

Instead of trying to solve the equation algebraically, we can check specific angles for solutions:

For x 0°:

cos0° 1 and sin30° 1/2 (Not a solution)

For x 30°:

cos60° 1/2 and sin60° √3/2 (Not a solution)

For x 60°:

cos120° -1/2 and sin90° 1 (Not a solution)

For x 90°:

cos180° -1 and sin120° √3/2 (Not a solution)

Step 6: Use numerical or graphical methods for approximate solutions

Given the complexity of the equation, numerical or graphical methods, such as the Newton-Raphson method or using a graphing calculator, may yield approximate solutions.

Conclusion

The equation cos2x sin x 30° can be solved through a combination of algebraic manipulation, checking specific angles, and employing numerical or graphical methods. If you need to find specific solutions or further assistance, feel free to ask!

Keywords: trigonometric equations, solving cos2x, sin x 30°