Question

How do you apply the double angle formula for $\sin 8x\cos 8x$?

Answer 1

In this problem we have to apply the double integration formula for the given trigonometric expression. We should know some trigonometric formulas and identities like the double integration formula to apply for it. We know that we have several double angle formulas in trigonometry, so we can take one of the formulas which relates the given trigonometric expression such as $\sin 2\theta =2\sin \theta \cos \theta$.

Complete step-by-step answer:
We know that the given trigonometric expression is,
$\sin 8x\cos 8x$
Here we have to apply the double integration formula.
We should know that double angle formulas are formulas expressing trigonometric functions of an angle in terms of functions of an angle.
We know that we have several double angle formulas in trigonometry, so we can take one of the formulas which relates the given trigonometric expression.
We can now take one of the double angle formulas which is related to the given expression,
$\sin 2\theta =2\sin \theta \cos \theta$
We can now write the above formula as,
$\Rightarrow \sin \theta \cos \theta =\dfrac{1}{2}\sin 2\theta$
We can see that the left-hand side of the above formula is similar to the given expression, so we can substitute $\theta =8x$ in the above formula, we get

& \Rightarrow \sin 8x\cos 8x=\dfrac{1}{2}\sin 2\left( 8x \right) \\\ & \Rightarrow \sin 8x\cos 8x=\dfrac{1}{2}\sin 16x \\\ \end{aligned}$$ Therefore, the answer is $$\sin 8x\cos 8x=\dfrac{1}{2}\sin 16x$$. **Note:** Students make mistakes while choosing a relevant double angle formula which is related to the given expression as we have several double angle formulas. We should also substitute the correct term in the formula to get the final answer correct as we have to substitute 8x in the formula instead of 8.