Solving the Equation: If sin x cos x and x is an Acute Angle, What is the Value of x?

When dealing with trigonometric equations, understanding the relationships between sine and cosine can significantly simplify the process. This article delves into the specific case where sin x cos x and x is an acute angle. By exploring traditional and alternative methods, we can accurately determine the value of x. This knowledge is crucial for students and professionals working in fields such as physics, engineering, and mathematics.

Introduction to Trigonometric Relationships

In trigonometry, the sine and cosine of an angle are fundamental concepts. The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the length of the hypotenuse. When these two ratios are equal, it implies a special relationship between the sides of a right triangle. This article will explore how to solve for the angle x given the equation sin x cos x, specifically when x is an acute angle.

Solving the Equation: Traditional Method

Let's start with a traditional approach to solving the equation sin x cos x. Since sin x represents the opposite side over the hypotenuse and cos x represents the adjacent side over the hypotenuse, the equation suggests that the opposite and adjacent sides are equal. For this to happen, the right triangle in question must be an isosceles right triangle, meaning the two acute angles are equal.

To find the measure of these angles, we can use the fact that the sum of the angles in a triangle is 180 degrees and that one of the angles is 90 degrees. Therefore, the remaining two angles must sum up to 90 degrees. Since these two angles are equal in an isosceles right triangle, each must be 45 degrees. Thus, the solution to sin x cos x for an acute angle x is:

x 45° or π/4 radians.

Explanation with Trigonometric Identities

A more formal approach involves using trigonometric identities. Recall the identity cos x sin(90° - x). Substituting this into the given equation, we get:

sin x sin(90° - x)

Using the identity sin α sin β, we have two possible solutions:

For the first equation:

x 90° - x 360°k

Solving for x, we get:

2x 90° 360°k

x 45° 180°k

This equation gives us angles that are 45 degrees plus multiples of 180 degrees, meaning the acute angle is x 45° 180°k. Since x is an acute angle, k must be zero, and we get x 45°.

The second equation:

x 180° - (90° - x) 360°k

Simplifies to:

x 90° x 360°k

This has no solution since 0 90° 360°k is not possible.

Alternative Approach: Using Tangent Function

Another method to solve the equation is to use the tangent function. The equation sin x cos x can be rewritten as:

sin x / cos x 1

This simplifies to:

tan x 1

The angle whose tangent is 1 is 45 degrees or π/4 radians. Therefore, the solution to the equation is:

x tan?1(1) 45° or π/4 radians.

Verification Using the Pythagorean Theorem

To further verify, consider a right triangle with x 45°. In a 45°-45°-90° triangle, the two legs are equal, and the hypotenuse is √2 times the length of each leg. If we assume the hypotenuse is 1, then each leg is 1/√2. The Pythagorean theorem confirms this:

(1/√2)2 (1/√2)2 1

Simplifying:

1/2 1/2 1

1 1

This confirms that x 45° is indeed a correct solution.

Conclusion

In conclusion, when dealing with the equation sin x cos x and finding the value of x for an acute angle, we find that the solution is x 45° or π/4 radians. This is achieved through various methods, including traditional geometric reasoning, trigonometric identities, and even the Pythagorean theorem. Understanding these concepts is fundamental in trigonometry and has wide applications in various fields of study.