Solving the Equation cos x sin 40° Using Trigonometric Identities

Trigonometric identities play a critical role in simplifying and solving trigonometric equations. One important identity is that the sine and cosine functions are cofunctions. This means that sin A cos (90° - A) and cos (90° - A) sin A. Let's explore how to solve the equation cos x sin 40° using these identities.

Step 1: Identifying the Cofunction Relationship

Utilizing the cofunction identity, we know that:

sin 40° cos (90° - 40°) cos 50°.

Therefore, if cos x sin 40°, we can rewrite this as:

cos x cos 50°

Step 2: Solving the Equation Using General Solutions

The general solution for the equation sin θ sin α is given by:

θ nπ ± α, where n is an integer.

Similarly, for cosine, the general solution is:

x 2πn ± π/2 - α, where n is an integer.

Applying these to our equation, we get:

cos x cos 50°

This implies that:

x 2πn ± 50°, where n is an integer.

For a general solution in degrees, we can write:

x n×360° ± 50°, where n is an integer.

Step 3: Simplifying the Solution

Based on the cofunction identity and general solution, we can directly find the solution as:

x 50°

Step 4: Using Inverse Cosine Function

We can also solve the equation cos x sin 40° by taking the inverse cosine of both sides. This gives us:

x arccos (sin 40°)

Since sin 40° 0.643, we get:

x arccos (0.643) ≈ 50°

Hence, we confirm that x 50° is the solution.

Understanding and applying these trigonometric identities and general solutions can help in solving more complex trigonometric equations. This method not only provides a clear and concise solution but also deepens our understanding of the relationships between trigonometric functions.