Trigonometry Examples

$\frac{28}{\sin (60)} = \frac{39}{\sin (A)}$

Step 1

Simplify $\frac{28}{\sin (60)}$ .

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Separate fractions.

$\frac{28}{1} \cdot \frac{1}{\sin (60)} = \frac{39}{\sin (A)}$

Convert from $\frac{1}{\sin (60)}$ to $\csc (60)$ .

$\frac{28}{1} \csc (60) = \frac{39}{\sin (A)}$

Divide $28$ by $1$ .

$28 \csc (60) = \frac{39}{\sin (A)}$

The exact value of $\csc (60)$ is $\frac{2}{\sqrt{3}}$ .

$28 \frac{2}{\sqrt{3}} = \frac{39}{\sin (A)}$

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

$28 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) = \frac{39}{\sin (A)}$

Combine and simplify the denominator.

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

$28 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} = \frac{39}{\sin (A)}$

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

$28 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} = \frac{39}{\sin (A)}$

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

$28 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} = \frac{39}{\sin (A)}$

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

$28 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} = \frac{39}{\sin (A)}$

Add $1$ and $1$ .

$28 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} = \frac{39}{\sin (A)}$

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

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

$28 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} = \frac{39}{\sin (A)}$

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

$28 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} = \frac{39}{\sin (A)}$

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

$28 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} = \frac{39}{\sin (A)}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$28 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} = \frac{39}{\sin (A)}$

Rewrite the expression.

$28 \frac{2 \sqrt{3}}{3^{1}} = \frac{39}{\sin (A)}$

Evaluate the exponent.

$28 \frac{2 \sqrt{3}}{3} = \frac{39}{\sin (A)}$

Multiply $28 \frac{2 \sqrt{3}}{3}$ .

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Combine $28$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{28 (2 \sqrt{3})}{3} = \frac{39}{\sin (A)}$

Multiply $2$ by $28$ .

$\frac{56 \sqrt{3}}{3} = \frac{39}{\sin (A)}$

Step 2

Simplify $\frac{39}{\sin (A)}$ .

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Separate fractions.

$\frac{56 \sqrt{3}}{3} = \frac{39}{1} \cdot \frac{1}{\sin (A)}$

Convert from $\frac{1}{\sin (A)}$ to $\csc (A)$ .

$\frac{56 \sqrt{3}}{3} = \frac{39}{1} \csc (A)$

Divide $39$ by $1$ .

$\frac{56 \sqrt{3}}{3} = 39 \csc (A)$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$A \approx 3.58006591 + 2 π n, 5.84471204 + 2 π n$ , for any integer $n$

Step 4