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Using Trigonometric Identities to Solve Tricky Trigonometric Expressions

March 11, 2025Technology2597
Using Trigonometric Identities to Solve Tricky Trigonometric Expressio

Using Trigonometric Identities to Solve Tricky Trigonometric Expressions

Are you trying to simplify or find the value of trigonometric expressions like sin3x cos3x using given conditions? In this article, we'll walk through a detailed example and explain the steps to solve exactly such a problem, combining various trigonometric identities. The example below will guide you through the process of finding the value of sin3x cos3x given that sin x cos x 1/4.

Given Problem

Find the value of sin3x cos3x given that sin x cos x 1/4.

Step-by-Step Solution

To solve this problem, we'll use an important trigonometric identity for the sum of cubes: a3 b3 (ab)(a2 - ab b2). In our case, let's set a sin x and b cos x. Thus, the identity becomes:

sin3x cos3x sin x cos x (sin2x - sin x cos x cos2x)

Step 1: Simplify using Pythagorean Identity

To further simplify the expression inside the parenthesis, we use the Pythagorean identity: sin2x cos2x 1. We know:

sin2x cos2x (sin x cos x)2

Substituting the given value (sin x cos x 1/4) into the Pythagorean identity, we have:

(sin x cos x)2 (1/4)2 1/16

Step 2: Solve for sin x cos x

Next, let's calculate sin x cos x. We use the square of the sum identity: (sin x cos x)2 sin2x cos2 x 1 2 sin x cos x. Substituting the value from the given, we find:

(1/4)2 1 2 sin x cos x

Solving for 2 sin x cos x yields:

2 sin x cos x 1/16 1 1/16 16/16 -15/16

Therefore, we have:

sin x cos x -15/32

Step 3: Substitute Back into the Original Expression

Now that we have the value of sin x cos x, we can substitute it back into the original expression to find the value of sin3x cos3x:

sin3x cos3x (1/4) (1 - (-15/32))

Final Calculation

Calculate the value inside the parenthesis:

1 - (-15/32) 32/32 15/32 47/32

Now, substitute back into the equation:

sin3x cos3x (1/4)(47/32) 47/128

General Case

In a more general case, if cos x sin x k, then the value of sin3x cos3x can be calculated as:

sin3x cos3x k/2 (3 - k2)

Conclusion

By understanding and utilizing trigonometric identities, we were able to find the value of a complex expression. This method and its similar application can be valuable for solving a variety of trigonometric problems, making it a useful skill for students and professionals.