/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 113 Verify the identity. $$\sin \f... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 113

Verify the identity. $$\sin \frac{\alpha}{3} \cos \frac{\alpha}{3}=\frac{1}{2} \sin \frac{2 \alpha}{3}$$

Short Answer

Expert verified
The given expression \(\sin \frac{\alpha}{3} \cos \frac{\alpha}{3}=\frac{1}{2} \sin \frac{2\alpha}{3}\) is verified using the double angle formula.

Step by step solution

01

Write down the Problem

Write down the given trigonometric identity: \(\sin \frac{\alpha}{3} \cos \frac{\alpha}{3}=\frac{1}{2} \sin \frac{2\alpha}{3}\).
02

Apply the double angle formula

We will use the double angle formula for sine, which says \(\sin 2x = 2\sin x \cos x\). In the given expression, we can think of \(x\) as \(\frac{\alpha}{3}\), which means \(\sin 2\frac{\alpha}{3} = 2\sin \frac{\alpha}{3} \cos \frac{\alpha}{3}\).
03

Compare the expressions

On comparing the new formula with the given trigonometric identity, we can see that both sides agree if we divide the double angle formula by \(2\). Thus, we have shown that \(\sin \frac{\alpha}{3} \cos \frac{\alpha}{3}=\frac{1}{2} \sin \frac{2\alpha}{3}\), verifying the given identity.

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