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Using Trigonometric Identities to Solve Tricky Trigonometric Expressions
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.