Finding tanx When sinx -2/5: A Comprehensive Guide

Trigonometry is a fundamental branch of mathematics that explores the relationships between angles and sides of triangles. One of the essential aspects of studying trigonometry is understanding the trigonometric identities and how they can be applied to solve complex problems. In this article, we will delve into the process of finding tanx given that sinx -2/5. We will cover the necessary steps, explanations, and visual aids to ensure a thorough understanding of the topic.

Understanding the Trigonometric Identity Involved

The relationship between the trigonometric functions sine, cosine, and tangent can be established through the following identity:

tanx sinx / cosx

This identity is crucial in solving for tanx when only the value of sinx is provided. However, to find cosx, we need to use another fundamental trigonometric identity:

sin2x cos2x 1

Deriving cosx from sinx

Given that sinx -2/5, we can substitute this value into our identity to find cosx. Let's go through the detailed steps:

Using the identity sin2x cos2x 1, substitute sinx -2/5:

sin2x (-2/5)2 4/25

cos2x 1 - sin2x 1 - 4/25 21/25

Therefore, cosx pm;√(21/25) pm;√21 / 5

Considering the Sign of Cosx

The sign of cosx depends on the quadrant where the angle x is located. Since sinx is negative, x is in either the third or fourth quadrant, where sine is negative:

Third quadrant: cosx is also negative. Fourth quadrant: cosx is positive.

Calculating tanx in Both Quadrants

Now we can find tanx for both cases:

Third quadrant (cosx -√21 / 5):

tanx sinx / cosx (-2/5) / (-√21 / 5) 2 / √21 2√21 / 21

Fourth quadrant (cosx √21 / 5):

tanx sinx / cosx (-2/5) / (√21 / 5) -2 / √21 -2√21 / 21

Conclusion

Given that sinx -2/5, the possible values for tanx are:

Third quadrant: tanx 2√21 / 21 Fourth quadrant: tanx -2√21 / 21

By understanding and applying the trigonometric identities, we can solve for tanx in various scenarios. The key is to consider the quadrant in which the angle lies to determine the sign of cosx, which then helps us find the correct value of tanx.

Additional Resources

If you want to deepen your understanding of trigonometry, explore the following resources:

Further reading on trigonometric identities and their applications in Trigonometry Cleared Up. Interactive tools to practice solving trigonometric equations at Trigonometry Practice.

By mastering these concepts and practicing regularly, you'll be able to tackle any trigonometric problem with confidence!