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Use a double-angle formula to rewrite the expression.7 − 14 sin2 x

Use a double-angle formula to rewrite the expression.7 − 14 sin2 x-example-1

2 Answers

3 votes

The expression
7-14sin^2(x) can be rewritten using the double-angle formula as cos(2x).

The double-angle formula for sine is sin(2θ)=2sin(θ)cos(θ)

To rewrite
7-14sin^2 (x) using the double-angle formula, we'll first express
sin^2 (x) in terms of the double-angle formula.


sin^2 (x)=
(1-cos(2x))/(2)​ (using the identity
sin^2 (x)=
(1-cos(2x))/(2)

So,
7-14sin^2(x) becomes:


7-14 * (1-cos(2x))/(2)

​Simplify that to get the expression in terms of cos(2x).

So, the expression
7-14sin^2(x) can be rewritten using the double-angle formula as cos(2x).

User Hansika
by
8.8k points
5 votes

7\cos \left(2x\right)

1) The point here is to use a double-angle formula, so let's pick this double-angle identity below:


1-2\sin ^2\left(x\right)=\cos \left(2x\right)

2) Now, let's rewrite that expression by making use of that expression above, and factoring it:


\begin{gathered} 7-14\sin^2\left(x\right)\:\:\: \\ \\ 7\cdot \:1-7\cdot \:2\sin ^2\left(x\right) \\ \\ 7\left(1-2\sin ^2\left(x\right)\right) \\ \\ \left(1-2\sin^2\left(x\right)\right)7\:\:\:Use\:this\:double\:angle\:identity:\:1-2\sin^2\left(x\right)=\cos\left(2x\right) \\ \\ (\cos(2x))7 \\ \\ 7\cos \left(2x\right) \end{gathered}

Note that we could simplify that since at the end, we could match it with the formula above.

User Manoj Gupta
by
8.4k points

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