Trigonometric Functions and Calculations Involving tan x 5/12

In trigonometry, it is essential to understand the relationship between the sine, cosine, and tangent of an angle in a right triangle. Given that tan x 5/12, we can calculate the values of sin x cos x using basic trigonometric identities and the Pythagorean theorem. This article guides you through this process and provides a deeper understanding of these trigonometric functions.

Understanding the Triangle

Given that tan x 5/12, we can interpret this ratio as the opposite side (5) and the adjacent side (12) of a right triangle. To find the hypotenuse, we use the Pythagorean theorem:

Hypotenuse √(52 122) √(25 144) √169 13.

Calculating sin x and cos x

Using the definitions of sine and cosine in terms of the sides of the right triangle:

sin x opposite/hypotenuse 5/13 cos x adjacent/hypotenuse 12/13

Therefore, the value of sin x and cos x is:

sin x 5/13 cos x 12/13

Now, to find the value of sin x cos x, we simply multiply:

sin x cos x 5/13 12/13 (5 12)/169 60/169 17/13.

Quadrant Consideration

When tan x 5/12, the angle x can belong to either the first or third quadrant. Therefore, the result for sin x cos x can be positive or negative:

sin x cos x 17/13 (positive) sin x cos x -17/13 (negative)

Thus, depending on the quadrant, the value of sin x cos x can be either 17/13 or -17/13.

Alternative Approach: Inverse Tangent

An alternative approach is to find the value of x using the inverse tangent function and then substitute it into the expression for sin x cos x using trigonometric identities:

x tan?1(5/12) ≈ 22.62°

Using the unit circle and the first quadrant:

sin x sin(22.62°) 5/13 cos x cos(22.62°) 12/13

Therefore:

sin x cos x (5/13) (12/13) 60/169 17/13.

Summary

In conclusion, given that tan x 5/12, the value of sin x cos x can be 17/13 or -17/13, depending on the quadrant. Using the definitions of sine and cosine in terms of the sides of a right triangle and the Pythagorean theorem, we can easily find the values and substitute them to get the final answer.

Understanding these concepts is crucial for solving more complex trigonometric problems. If you need further assistance, feel free to explore more detailed explanations and examples provided in advanced trigonometry resources.