How to Determine the Value of tan3A When tanA 4/3

Understanding how to determine the value of tan3A when given tanA 4/3 is a fundamental skill in trigonometry. This article will provide a detailed explanation of the process along with a clear example.

Understanding the Problem

Given that tanA 4/3, we want to find the value of tan3A. We will use the tangent triple angle formula to solve this problem:

The Tangent Triple Angle Formula

The tangent triple angle formula is:

tan3A (3tanA - tan3A) / (1 - 3tan2A)

Deriving the Value of tan3A Step-by-Step

tanA 4/3

tan2A (4/3)2 16/9

tan3A (4/3)3 64/27

tan3A (3 * (4/3) - (64/27)) / (1 - 3 * (16/9))

Numerator: 3 * (4/3) - (64/27) 4 - 64/27 (4 * 27) - 64 108 - 64 44/27

tan3A (44/27) / (-117/27) -44/117

Verification and Simplification

Verification of the result is crucial:

Using the tangent triple angle formula:

tan3A (3 * (4/3) - (4/3)3) / (1 - 3 * (4/3)2)

tan3A (4 - 64/27) / (1 - 48/9)

tan3A (108 - 64) / -39

tan3A -44/117

Summary

The value of tan3A when tanA 4/3 is -44/117.

Conclusion

This method can be applied to a wide range of problems involving the tangent triple angle formula. Understanding these steps ensures a robust grasp of trigonometric identities and their practical applications.