Trigonometry Examples

$\frac{5 \sin (30)}{\sin (60)}$

Step 1

Separate fractions.

$\frac{5}{1} \cdot \frac{\sin (30)}{\sin (60)}$

Step 2

Rewrite $\frac{\sin (30)}{\sin (60)}$ as a product.

$\frac{5}{1} (\sin (30) \frac{1}{\sin (60)})$

Step 3

Write $\sin (30)$ as a fraction with denominator $1$ .

$\frac{5}{1} (\frac{\sin (30)}{1} \cdot \frac{1}{\sin (60)})$

Step 4

Simplify.

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Step 4.1

Divide $\sin (30)$ by $1$ .

$\frac{5}{1} (\sin (30) \frac{1}{\sin (60)})$

Step 4.2

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

$\frac{5}{1} (\sin (30) \csc (60))$

Step 5

Divide $5$ by $1$ .

$5 (\sin (30) \csc (60))$

Step 6

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$5 (\frac{1}{2} \csc (60))$

Step 7

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

$5 (\frac{1}{2} \cdot \frac{2}{\sqrt{3}})$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Cancel the common factor.

$5 (\frac{1}{2} \cdot \frac{2}{\sqrt{3}})$

Step 8.2

Rewrite the expression.

$5 \frac{1}{\sqrt{3}}$

Step 9

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

$5 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 10

Combine and simplify the denominator.

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Step 10.1

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

$5 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 10.2

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

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 10.3

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

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 10.4

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

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 10.5

Add $1$ and $1$ .

$5 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 10.6

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

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Step 10.6.1

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

$5 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 10.6.2

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

$5 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 10.6.3

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

$5 \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$5 \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 10.6.4.2

Rewrite the expression.

$5 \frac{\sqrt{3}}{3^{1}}$

Step 10.6.5

Evaluate the exponent.

$5 \frac{\sqrt{3}}{3}$

Step 11

Combine $5$ and $\frac{\sqrt{3}}{3}$ .

$\frac{5 \sqrt{3}}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{3}}{3}$

Decimal Form:

$2.88675134 \dots$