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QuestionFind sin⁡θ\sin \theta given csc⁡θ=184\csc \theta = \frac{\sqrt{18}}{4}. Simplify and rationalize if needed.

Studdy Solution

STEP 1

Assumptions1. The given value is csc⁡θ=184\csc \theta=\frac{\sqrt{18}}{4}. . We need to find the exact value of sin⁡θ\sin \theta using the appropriate reciprocal identity.
3. The reciprocal identity of csc⁡θ\csc \theta is sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}.
4. We need to rationalize denominators when applicable.

STEP 2

We will 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=118\sin \theta = \frac{1}{\frac{\sqrt{18}}{}}

STEP 4

implify the expression.
sin⁡θ=418\sin \theta = \frac{4}{\sqrt{18}}

STEP 5

Rationalize the denominator by multiplying the numerator and the denominator by 18\sqrt{18}.
sin⁡θ=41818\sin \theta = \frac{4\sqrt{18}}{18}

STEP 6

implify the radical in the numerator by finding the square root of18.
sin⁡θ=49⋅218=4⋅3218\sin \theta = \frac{4\sqrt{9\cdot2}}{18} = \frac{4\cdot3\sqrt{2}}{18}

STEP 7

implify the fraction.
sin⁡θ=223\sin \theta = \frac{2\sqrt{2}}{3}So, the exact value of sin⁡θ\sin \theta is 223\frac{2\sqrt{2}}{3}.

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