Understanding Trigonometric Identities: The Value of cos2x Given sinx 3/5

Trigonometry is a fundamental branch of mathematics, dealing with the relationships between the angles and the sides of triangles. One of the primary aspects of trigonometry is understanding the relationships between different trigonometric functions, particularly through the use of trigonometric identities. In this article, we explore a specific example of finding cos2x given that sinx 3/5. We will use a combination of basic trigonometric identities to solve this problem.

The Trig Identity: cos2x 1 - 2sin2x

The identity cos2x 1 - 2sin2x is a fundamental tool in solving trigonometric problems. Let's walk through the process step by step using a specific example.

Given:

sinx 3/5

We need to find the value of cos2x.

Applying the identity:

First, we need to square the sine value:

sin2x (3/5)2 9/25

Substitute into the identity:

cos2x 1 - 2sin2x 1 - 2(9/25) 25/25 - 18/25 7/25

Hence, the value of cos2x is 7/25.

Alternative Methods to Verify the Result

Giving an extra insight, here are a few alternative methods to verify the result:

We can calculate cosx using the Pythagorean identity:

cosx √1 - sin2x √1 - (9/25) 4/5

Now, using the double angle formula for sine: sin2x 2sinxcosx

sin2x 2(3/5)(4/5) 24/25

Alternatively, using the cosine double angle formula: cos2x cos2x - sin2x

cos2x (16/25) - (9/25) 7/25

Conclusion

In conclusion, understanding and applying trigonometric identities is crucial in solving a variety of problems in mathematics and physics. The value of cos2x when sinx 3/5 can be found using the identity cos2x 1 - 2sin2x, and we have provided various methods to double-check the result. This example showcases the importance of these identities in simplifying and solving trigonometric expressions.

Keywords:

trigonometric identities, sinx, cos2x