$$ -90^{\circ} \leq heta \leq 90^{\circ} $$. Find the value of $$ heta $$ in degrees. $$ \csc ( heta)=\frac{2 \sqrt{3}}{3} $$ $$ heta= $$ Submit

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$$ -90^{\circ} \leq 	heta \leq 90^{\circ} $$. Find the value of $$ 	heta $$ in degrees. $$ \csc (	heta)=\frac{2 \sqrt{3}}{3} $$ $$ 	heta= $$ Submit

UpStudy AI · Accepted Answer

To find the value of $$ \theta $$ given that $$ \csc(\theta) = \frac{2\sqrt{3}}{3} $$, we first recall the relationship between cosecant and sine:

$$
\csc(\theta) = \frac{1}{\sin(\theta)}
$$

Thus, we can express sine in terms of cosecant:

$$
\sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{\frac{2\sqrt{3}}{3}} = \frac{3}{2\sqrt{3}}
$$

To simplify $$ \frac{3}{2\sqrt{3}} $$, we can multiply the numerator and the denominator by $$ \sqrt{3} $$:

$$
\sin(\theta) = \frac{3\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2}
$$

Next, we need to find the angle $$ \theta $$ such that:

$$
\sin(\theta) = \frac{\sqrt{3}}{2}
$$

The sine function equals $$ \frac{\sqrt{3}}{2} $$ at the angles $$ 60^\circ $$ and $$ 120^\circ $$. However, since we are restricted to the interval $$ -90^\circ \leq \theta \leq 90^\circ $$, we only consider $$ 60^\circ $$.

Thus, the value of $$ \theta $$ is:

$$
\theta = 60^\circ
$$

Therefore, the final answer is:

$$
\boxed{60}
$$
$$ \theta = 60^\circ $$

\( -90^{\circ} \leq \theta \leq 90^{\circ} \). Find the value of \( \theta \) in degrees. \( \csc (\theta)=\frac{2 \sqrt{3}}{3} \) \( \theta= \) Submit

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