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QuestionFind sin⁡θ\sin \theta given csc⁡θ=1176\csc \theta = \frac{\sqrt{117}}{6}. Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. We are given the value of csc⁡θ=1176\csc \theta = \frac{\sqrt{117}}{6} . We need to find the value of sin⁡θ\sin \theta
3. We know the reciprocal identity sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

STEP 2

We will use the reciprocal identity to find the value of sin⁡θ\sin \theta. The reciprocal identity issin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

STEP 3

Now, plug in the given value for csc⁡θ\csc \theta into the reciprocal identity to calculate sin⁡θ\sin \theta.
sin⁡θ=11176\sin \theta = \frac{1}{\frac{\sqrt{117}}{6}}

STEP 4

To simplify the fraction, we can multiply the numerator and the denominator by the denominator of the fraction in the denominator.
sin⁡θ=1×6117\sin \theta = \frac{1 \times6}{\sqrt{117}}

STEP 5

Calculate the value of sin⁡θ\sin \theta.
sin⁡θ=117\sin \theta = \frac{}{\sqrt{117}}

STEP 6

To rationalize the denominator, we multiply the numerator and the denominator by 117\sqrt{117}.
sin⁡θ=6×117117\sin \theta = \frac{6 \times \sqrt{117}}{117}

STEP 7

implify the fraction to find the exact value of sin⁡θ\sin \theta.
sin⁡θ=6117117\sin \theta = \frac{6\sqrt{117}}{117}The exact value of sin⁡θ\sin \theta is 6117117\frac{6\sqrt{117}}{117}.

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