Dive into the fascinating world of Sine Cosine Derivation! We're going to unravel the trigonometric identity ##sin(x)cos(x) = (1/2)sin(2x)##. This Sine Cosine Derivation will show you how to transform a product of sine and cosine functions into a simpler form involving a sine function with a doubled angle. It's a fundamental concept in trigonometry, and we'll make it easy to understand.
This derivation relies on a powerful tool: the product-to-sum trigonometric identity. By applying this identity, we'll be able to transform the product into a sum of sine functions. Following a few simple steps, we'll arrive at the desired result, demonstrating the elegance and power of trigonometric identities. Let's embark on this Sine Cosine Derivation journey together!
Derivation of ## \sin(x) \cos(x) ##
Problem Statement
Derive the trigonometric identity:
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\sin(x) \cos(x) = \frac{1}{2} \sin(2x)
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This derivation utilizes the product-to-sum trigonometric identities.
Solution
Understanding the Problem
We are asked to show that the product of sine and cosine functions can be expressed as a sine function of a multiple angle. This is a fundamental trigonometric identity.
Solving the Problem
Step 1: Recall the Product-to-Sum Identity
A key trigonometric identity is the product-to-sum formula for sine and cosine:
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\sin(A) \cos(B) = \frac{1}{2} [\sin(A + B) + \sin(A - B)]
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Step 2: Apply the Identity
In our case, ## A = x ## and ## B = x ##. Substituting these values into the product-to-sum formula gives:
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\sin(x) \cos(x) = \frac{1}{2} [\sin(x + x) + \sin(x - x)]
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Step 3: Simplify the Arguments
Simplifying the arguments of the sine functions yields:
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\sin(x + x) = \sin(2x), \quad \sin(x - x) = \sin(0) = 0
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Step 4: Combine the Results
Substituting the simplified results back into the equation gives:
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\sin(x) \cos(x) = \frac{1}{2} [\sin(2x) + \sin(0)]
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Step 5: Final Simplification
Since ## \sin(0) = 0 ##, the equation simplifies to:
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\sin(x) \cos(x) = \frac{1}{2} \sin(2x)
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Final Solution
Therefore, the derived identity is:
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\sin(x) \cos(x) = \frac{1}{2} \sin(2x)
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This derivation showcases the application of product-to-sum trigonometric identities to simplify a trigonometric expression. These identities are crucial for various mathematical and scientific applications.
| Step | Equation/Formula | Explanation | Relevant Concepts |
|---|---|---|---|
| 1 | ###\sin(A) \cos(B) = \frac{1}{2} [\sin(A + B) + \sin(A - B)]### | Product-to-sum identity for sine and cosine. | Trigonometric Identities, Sine Cosine Derivation |
| 2 | ###\sin(x) \cos(x) = \frac{1}{2} [\sin(x + x) + \sin(x - x)]### | Substituting A = x and B = x into the product-to-sum identity. | Substitution, Trigonometric Identities |
| 3 | ###\sin(x + x) = \sin(2x), \quad \sin(x - x) = \sin(0) = 0### | Simplifying the arguments of the sine functions. | Trigonometric Function Simplification |
| 4 | ###\sin(x) \cos(x) = \frac{1}{2} [\sin(2x) + \sin(0)]### | Substituting the simplified results back into the equation. | Substitution, Trigonometric Identities |
| 5 | ###\sin(x) \cos(x) = \frac{1}{2} \sin(2x)### | Simplifying the equation by evaluating ## \sin(0) = 0 ##. | Simplification, Trigonometric Identities, Sine Cosine Derivation |
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