Trigonometry Examples

$\frac{11.5}{\sin (Z)} = \frac{13.8}{\sin (60 °)}$

Step 1

Factor each term.

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Step 1.1

Separate fractions.

$\frac{11.5}{\sin (Z)} = \frac{13.8}{1} \cdot \frac{1}{\sin (60 °)}$

Step 1.2

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

$\frac{11.5}{\sin (Z)} = \frac{13.8}{1} \csc (60 °)$

Step 1.3

Divide $13.8$ by $1$ .

$\frac{11.5}{\sin (Z)} = 13.8 \csc (60 °)$

Step 1.4

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2}{\sqrt{3}}$

Step 1.5

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

$\frac{11.5}{\sin (Z)} = 13.8 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.6.2

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.6.3

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.6.4

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.6.5

Add $1$ and $1$ .

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.6.6

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

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Step 1.6.6.1

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.6.6.2

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.6.6.3

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

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.6.6.4.2

Rewrite the expression.

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{3^{1}}$

Step 1.6.6.5

Evaluate the exponent.

$\frac{11.5}{\sin (Z)} = 13.8 \frac{2 \sqrt{3}}{3}$

Step 1.7

Multiply $13.8 \frac{2 \sqrt{3}}{3}$ .

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Step 1.7.1

Combine $13.8$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{11.5}{\sin (Z)} = \frac{13.8 (2 \sqrt{3})}{3}$

Step 1.7.2

Multiply $2$ by $13.8$ .

$\frac{11.5}{\sin (Z)} = \frac{27.6 \sqrt{3}}{3}$

Step 1.7.3

Multiply $27.6$ by $\sqrt{3}$ .

$\frac{11.5}{\sin (Z)} = \frac{47.80460228}{3}$

Step 1.8

Divide $47.80460228$ by $3$ .

$\frac{11.5}{\sin (Z)} = 15.93486742$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\sin (Z), 1$

Step 2.2

The LCM of one and any expression is the expression.

$\sin (Z)$

Step 3

Multiply each term in $\frac{11.5}{\sin (Z)} = 15.93486742$ by $\sin (Z)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{11.5}{\sin (Z)} = 15.93486742$ by $\sin (Z)$ .

$\frac{11.5}{\sin (Z)} \sin (Z) = 15.93486742 \sin (Z)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\sin (Z)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{11.5}{\sin (Z)} \sin (Z) = 15.93486742 \sin (Z)$

Step 3.2.1.2

Rewrite the expression.

$11.5 = 15.93486742 \sin (Z)$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $15.93486742 \sin (Z) = 11.5$ .

$15.93486742 \sin (Z) = 11.5$

Step 4.2

Divide each term in $15.93486742 \sin (Z) = 11.5$ by $15.93486742$ and simplify.

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Step 4.2.1

Divide each term in $15.93486742 \sin (Z) = 11.5$ by $15.93486742$ .

$\frac{15.93486742 \sin (Z)}{15.93486742} = \frac{11.5}{15.93486742}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $15.93486742$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{15.93486742 \sin (Z)}{15.93486742} = \frac{11.5}{15.93486742}$

Step 4.2.2.1.2

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

$\sin (Z) = \frac{11.5}{15.93486742}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $11.5$ by $15.93486742$ .

$\sin (Z) = 0.72168783$

Step 5

Take the inverse sine of both sides of the equation to extract $Z$ from inside the sine.

$Z = \arcsin (0.72168783)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arcsin (0.72168783)$ .

$Z = 46.19400773$

Step 7

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$Z = 180 - 46.19400773$

Step 8

Subtract $46.19400773$ from $180$ .

$Z = 133.80599226$

Step 9

Find the period of $\sin (Z)$ .

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Step 9.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 9.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 9.4

Divide $360$ by $1$ .

$360$

Step 10

The period of the $\sin (Z)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$Z = 46.19400773 + 360 n, 133.80599226 + 360 n$ , for any integer $n$