simplify the expression by using a double - angle formula or a half - angle formula.\n(a) ( 2 sin ( 33 ^ {…

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

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

Answer

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))