Trigonometry Examples

$\csc (θ) = \frac{2 \sqrt{3}}{3}$

Step 1

To find the value of $\sin (θ)$ , use the fact that $\frac{1}{\csc (θ)}$ then substitute in the known values.

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

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Multiply $\frac{3}{2 \sqrt{3}}$ by $1$ .

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

Step 2.3

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply $\frac{3}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.4.2

Move $\sqrt{3}$ .

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

Step 2.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.4.4

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = \frac{1}{\csc (θ)} = \frac{3 \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 2.4.6

Add $1$ and $1$ .

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

Step 2.4.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (θ) = \frac{1}{\csc (θ)} = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (θ) = \frac{1}{\csc (θ)} = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 2.4.7.4

Cancel the common factor of $2$ .

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Step 2.4.7.4.1

Cancel the common factor.

$\sin (θ) = \frac{1}{\csc (θ)} = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 2.4.7.4.2

Rewrite the expression.

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

Step 2.4.7.5

Evaluate the exponent.

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

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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