Solving the Trigonometric Equation: If sin 3x - 23° sec 5x - 37° 1, Then What is x?

This article delves into the process of solving a specific trigonometric equation, providing a step-by-step explanation and the final result. We explore the underlying trigonometric identities and principles to arrive at a solution for x.

Understanding Trigonometric Equations

Trigonometric equations involve trigonometric functions such as sine (sin), cosine (cos), and secant (sec). These functions depend on the angle x, and the equation here is a bit more complex, involving both sine and secant functions. The secant function is the reciprocal of the cosine function, so sec(5x - 37°) can be represented as 1/cos(5x - 37°).

Given Equation and Its Transformation

The given equation is:

sin 3x - 23° sec 5x - 37° 1

To solve this, let's first understand the relationship between sine and cosine using a complementary angle identity. Specifically, we know that:

sin(90° - θ) cosθ

And similarly, for secant:

sec(90° - θ) cos(90° - θ) sinθ

Simplifying the Equation

Using these identities, we can rewrite the given equation to:

sin 3x - 23° cos(5x - 37°)

This simplifies further using the complementary angle identity:

sin 3x - 23° cos(90° - (5x - 37°))

Which simplifies to:

sin 3x - 23° cos(90° - 5x 37°)

Further simplifying:

sin 3x - 23° cos(127° - 5x)

Setting Up the Equation for x

Now, let's set up the equation where the arguments of the sine and cosine functions are equal to 90°, as the expression will equate to 1 only if it is equal to a right angle:

3x - 23° 127° - 5x

Let's solve for x:

3x 5x 127° 23°

8x 150°

x 18.75°

Verification Process

To verify the solution, let's substitute x 18.75° back into the original equation:

3x - 23° 3(18.75°) - 23° 56.25° - 23° 33.25°

5x - 37° 5(18.75°) - 37° 93.75° - 37° 56.75°

Now, let's find the sine and cosine values for these angles:

sin 33.25° 0.5483

cos 56.75° 0.5483

Given expression:

sin 33.25° cos 56.75° 0.5483 0.5483 1

Thus, the given expression equals 1, confirming our solution.

Conclusion

In conclusion, the value of x that satisfies the equation sin 3x - 23° sec 5x - 37° 1 is 18.75°. This solution was derived through the application of trigonometric identities and the verification process to ensure the correctness of the result.