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QuestionFind sin⁡θ\sin \theta given csc⁡θ=204\csc \theta = \frac{\sqrt{20}}{4}. Simplify your answer, including any radicals.

Studdy Solution

STEP 1

Assumptions1. The given value is for csc⁡θ\csc \theta which is 204\frac{\sqrt{20}}{4} . We need to find the value of sin⁡θ\sin \theta
3. We are using the reciprocal identity sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

STEP 2

We can use the reciprocal identity 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 find the value of sin⁡θ\sin \theta.
sin⁡θ=120\sin \theta = \frac{1}{\frac{\sqrt{20}}{}}

STEP 4

implify the expression by multiplying the numerator and the denominator by4.
sin⁡θ=420\sin \theta = \frac{4}{\sqrt{20}}

STEP 5

Rationalize the denominator by multiplying the numerator and the denominator by 20\sqrt{20}.
sin⁡θ=42020\sin \theta = \frac{4\sqrt{20}}{20}

STEP 6

implify the expression by dividing the numerator and the denominator by4.
sin⁡θ=205\sin \theta = \frac{\sqrt{20}}{5}The exact value of sin⁡θ\sin \theta is 205\frac{\sqrt{20}}{5}.

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