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QuestionFind sin⁡θ\sin \theta given csc⁡θ=442\csc \theta = \frac{\sqrt{44}}{2}. Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. The given value is csc⁡θ=44\csc \theta=\frac{\sqrt{44}}{} . We need to find the value of sin⁡θ\sin \theta using the reciprocal identity of csc⁡θ\csc \theta
3. The reciprocal identity of csc⁡θ\csc \theta is sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

STEP 2

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

STEP 3

Now, plug in the given value for csc⁡θ\csc \theta to calculate sin⁡θ\sin \theta.
sin⁡θ=1442\sin \theta = \frac{1}{\frac{\sqrt{44}}{2}}

STEP 4

To simplify the expression, we can multiply the numerator and the denominator by2.
sin⁡θ=244\sin \theta = \frac{2}{\sqrt{44}}

STEP 5

Rationalize the denominator by multiplying the numerator and the denominator by 44\sqrt{44}.
sin⁡θ=24444\sin \theta = \frac{2\sqrt{44}}{44}

STEP 6

implify the fraction by dividing the numerator and the denominator by2.
sin⁡θ=4422\sin \theta = \frac{\sqrt{44}}{22}So, the exact value of sin⁡θ\sin \theta is 4422\frac{\sqrt{44}}{22}.

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