Math

QuestionFind the exact value of sin132\sin^{-1} \frac{\sqrt{3}}{2}.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the exact value of the expression sin13\sin ^{-1} \frac{\sqrt{3}}{}. The function sin1\sin ^{-1} is the inverse sine function, also known as arcsine.
3. The range of the arcsine function is between -90 degrees and90 degrees (or between -π/ and π/ in radians).
4. We are looking for an angle whose sine is 3\frac{\sqrt{3}}{}.

STEP 2

We need to find an angle in the range of the arcsine function whose sine is 2\frac{\sqrt{}}{2}. We can do this by recalling the unit circle or the special right triangles (30-60-90 or45-45-90).

STEP 3

From the unit circle or the special right triangles, we know that sin(60)=32\sin(60^{\circ}) = \frac{\sqrt{3}}{2}.

STEP 4

Therefore, the exact value of the expression sin132\sin ^{-1} \frac{\sqrt{3}}{2} is 6060^{\circ}.If the answer is required in radians, we can convert the degrees to radians.

STEP 5

The conversion from degrees to radians is done by multiplying the degree measure by π180\frac{\pi}{180}.60=60×π180=π360^{\circ} =60 \times \frac{\pi}{180} = \frac{\pi}{3}So, the exact value of the expression sin132\sin ^{-1} \frac{\sqrt{3}}{2} is π3\frac{\pi}{3} in radians.
The exact value of the expression sin132\sin ^{-1} \frac{\sqrt{3}}{2} is 6060^{\circ} or π3\frac{\pi}{3} radians.

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