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.