Question

simplify the expression by using a double - angle formula or a half - angle formula.
(a) ( 2 sin ( 33 ^ { circ } ) cos ( 33 ^ { circ } ) )
( sin ( 66 ^ { circ } ) )
amazing work.
(b) ( 2 sin ( 7 \theta ) cos ( 7 \theta ) )
( sin ( 16 ) )

Explanation:

Step1: Recall the double - angle formula for sine

The double - angle formula for sine is \(\sin(2\alpha)=2\sin\alpha\cos\alpha\).

Step2: Identify \(\alpha\) in the given expression

For the expression \(2\sin(7\theta)\cos(7\theta)\), we can see that \(\alpha = 7\theta\).

Step3: Apply the double - angle formula

Substituting \(\alpha = 7\theta\) into the formula \(\sin(2\alpha)=2\sin\alpha\cos\alpha\), we get \(2\sin(7\theta)\cos(7\theta)=\sin(2\times7\theta)=\sin(14\theta)\).

Answer:

\(\sin(14\theta)\)