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QuestionFind θ\theta in [0∘,360∘)[0^{\circ}, 360^{\circ}) such that csc⁡θ=233\csc \theta=\frac{2 \sqrt{3}}{3}. θ=□∘\theta=\square^{\circ}

Studdy Solution

STEP 1

Assumptions1. The function value is given as csc⁡θ=33\csc \theta=\frac{ \sqrt{3}}{3} . The range of θ\theta is from 0∘0^{\circ} to 360∘360^{\circ} (inclusive of 0∘0^{\circ}, exclusive of 360∘360^{\circ})
3. We are looking for all possible values of θ\theta within this range.

STEP 2

The cosecant function, csc⁡θ\csc \theta, is the reciprocal of the sine function, sin⁡θ\sin \theta. Therefore, we can rewrite the given equation assin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

STEP 3

Substitute the given value of csc⁡θ\csc \theta into the equationsin⁡θ=1233\sin \theta = \frac{1}{\frac{2 \sqrt{3}}{3}}

STEP 4

implify the right side of the equationsin⁡θ=323\sin \theta = \frac{3}{2 \sqrt{3}}

STEP 5

Rationalize the denominator of the right side of the equationsin⁡θ=33=32\sin \theta = \frac{3 \sqrt{3}}{} = \frac{\sqrt{3}}{2}

STEP 6

Now, we need to find all angles θ\theta in the interval [0∘,360∘)[0^{\circ},360^{\circ}) for which sin⁡θ=32\sin \theta = \frac{\sqrt{3}}{2}.

STEP 7

From the unit circle or the sine function graph, we know that sin⁡θ=32\sin \theta = \frac{\sqrt{3}}{2} for θ=60∘\theta =60^{\circ} and θ=120∘\theta =120^{\circ} within the given interval.
Therefore, the solution to the equation is θ=60∘,120∘\theta =60^{\circ},120^{\circ}.

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