Solving Trigonometric Equations: sinx - 10 cosx 10

In the realm of trigonometry, equations involving sine and cosine functions can be rather intriguing and challenging. This article explores the solution to the equation sinx - 10 cosx 10 and provides insights into the underlying trigonometric identities and solution techniques.

Understanding the Problem

The given equation is sin x - 10 cos x 10. To solve this equation, we can utilize the trigonometric identity sin θ cos(90° - θ). This identity will help us transform the equation into a more manageable form.

Transforming the Equation

Using the identity, we set x - 10 90° - x 10. This transformation allows us to derive a simpler equation in terms of x.

Step-by-Step Solution

Step 1: Expand the equation:

(x - 10 90° - x - 10)

Step 2: Simplify the equation:

(x - 10 80° - x)

Step 3: Add x to both sides to combine like terms:

(2x - 10 80°)

Step 4: Add 10 to both sides to isolate the term with x:

(2x 90°)

Step 5: Divide both sides by 2 to solve for x:

(x 45°)

This seems to be the primary solution, but let's explore further to ensure there are no other solutions.

Periodic Solutions

Given the periodic nature of the sine and cosine functions, we can consider the general solution:

(sin x - 10 cos x 10 implies x - 10 90° - x 10 360° k for some integer k.

Substituting the simplified form, we get:

(x - 10 90° - x 10 360° k

This leads to the same equation derived earlier, providing no additional solutions.

Alternative Approach

To further validate the solution, we can utilize the cofunction identity: sin 90° - x cos x. Using this, the equation can be rewritten as:

(sin x - 10 sin 90° - x - 10

Which simplifies to:

(x - 10 90° - x - 10 k 360°

Solving this, we find:

(x - 10 90° - x - 10 k 360°

This again leads to the same solution:

(x 45°

Therefore, the only solution to the equation is x 45°.

Conclusion

In summary, we have derived that the equation sin x - 10 cos x 10 has the primary solution x 45°. This solution is consistent with the periodic properties of the sine and cosine functions and the application of trigonometric identities.

Equations: x 45°