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QuestionFind sin⁡θ\sin \theta given csc⁡θ=284\csc \theta = \frac{\sqrt{28}}{4}. Rationalize denominators if needed.

Studdy Solution

STEP 1

Assumptions1. The given value is csc⁡θ=284\csc \theta = \frac{\sqrt{28}}{4} . We need to find the value of sin⁡θ\sin \theta
3. We know that sin⁡θ\sin \theta and csc⁡θ\csc \theta are reciprocal identities, i.e., sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

STEP 2

We can use the reciprocal identity of sin⁡θ\sin \theta and 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=128\sin \theta = \frac{1}{\frac{\sqrt{28}}{}}

STEP 4

This is a complex fraction, we can simplify it by multiplying the numerator and the denominator of the big fraction by the reciprocal of the small fraction.
sin⁡θ=1×428\sin \theta =1 \times \frac{4}{\sqrt{28}}

STEP 5

implify the multiplication.
sin⁡θ=428\sin \theta = \frac{4}{\sqrt{28}}

STEP 6

Rationalize the denominator by multiplying the numerator and the denominator by 28\sqrt{28}.
sin⁡θ=4×2828\sin \theta = \frac{4 \times \sqrt{28}}{28}

STEP 7

implify the fraction.
sin⁡θ=42828\sin \theta = \frac{4\sqrt{28}}{28}

STEP 8

We can simplify the 28\sqrt{28} to 272\sqrt{7}, and the fraction to 27\frac{2}{7}.
sin⁡θ=277\sin \theta = \frac{2\sqrt{7}}{7}So, the exact value of sin⁡θ\sin \theta for the given value of csc⁡θ\csc \theta is 277\frac{2\sqrt{7}}{7}.

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