Solving Trigonometric Expressions with Given Tangent Value: A Comprehensive Guide

Trigonometry is a fundamental branch of mathematics dealing with the relationships between the angles and sides of triangles. One of the most common tasks in trigonometry is to solve expressions involving trigonometric functions given certain conditions. In this article, we will explore a specific case where tan A 4/5. Our goal is to find the value of the expression (2sin A - 3cos A) / (2sin A 3cos A).

Step-by-Step Solution

To solve the expression (2sin A - 3cos A) / (2sin A 3cos A) given that tan A 4/5, we can follow these steps:

Understand the tangent relationship: Given tan A 4/5, we know that tan A sin A / cos A. Solve for sin A and cos A: We set sin A 4k and cos A 5k. Here, k is a constant to be determined. To find k, we use the Pythagorean identity: sin^2 A cos^2 A 1. Substitute the identities: The Pythagorean identity gives us 4k^2 5k^2 1. This simplifies to 41k^2 1, so k^2 1/41, which means k 1/√41. Calculate sin A and cos A: Substituting back, we get sin A 4k 4/√41 and cos A 5k 5/√41. Substitute into the expression: The numerator of the expression becomes: 2sin A - 3cos A 2(4/√41) - 3(5/√41) (8 - 15)/√41 -7/√41. The denominator of the expression becomes: 2sin A 3cos A 2(4/√41) 3(5/√41) (8 15)/√41 23/√41. The final expression simplifies to: (-7/√41) / (23/√41) -7/23.

Summary

By following these steps, we have determined that the value of the expression (2sin A - 3cos A) / (2sin A 3cos A) is -7/23 when tan A 4/5.

Additional Resources

For a deeper understanding of trigonometric identities and related mathematical concepts, we recommend exploring the following resources:

Mathworld () Wikipedia () Khan Academy ()

These resources provide extensive guidance and examples to help you master trigonometric expressions and identities.