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QuestionSimplify 4−x2\sqrt{4-x^{2}} using the substitution x=2sin⁡θx=2 \sin \theta.

Studdy Solution

STEP 1

Assumptions1. The given expression is 4−x\sqrt{4-x^{}} . The substitution to be made is x=sin⁡θx= \sin \theta
3. θ\theta is a real number

STEP 2

Substitute x=2sin⁡θx=2 \sin \theta into the expression.
4−(2sin⁡θ)2\sqrt{4-(2 \sin \theta)^{2}}

STEP 3

implify the expression inside the square root.
−sin⁡2θ\sqrt{- \sin^{2} \theta}

STEP 4

Factor out the common factor of4 from the expression inside the square root.
4(1−sin⁡2θ)\sqrt{4(1-\sin^{2} \theta)}

STEP 5

Use the Pythagorean identity 1−sin⁡2θ=cos⁡2θ1-\sin^{2}\theta=\cos^{2}\theta to simplify the expression inside the square root.
4cos⁡2θ\sqrt{4\cos^{2} \theta}

STEP 6

Take the square root of the expression inside the square root.
2∣cos⁡θ∣2|\cos \theta|

STEP 7

Since θ\theta is a real number, the absolute value can be removed, leaving the final simplified expression.
2cos⁡θ2\cos \thetaThe simplified expression is 2cos⁡θ2\cos \theta.

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