Trigonometry Examples

$\frac{11}{\sin (77)} = \frac{x}{\sin (45)}$

Step 1

Rewrite the equation as $\frac{x}{\sin (45)} = \frac{11}{\sin (77)}$ .

$\frac{x}{\sin (45)} = \frac{11}{\sin (77)}$

Step 2

Multiply both sides of the equation by $\sin (45)$ .

$\sin (45) \frac{x}{\sin (45)} = \sin (45) \frac{11}{\sin (77)}$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $\sin (45)$ .

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Step 3.1.1.1

Cancel the common factor.

$\sin (45) \frac{x}{\sin (45)} = \sin (45) \frac{11}{\sin (77)}$

Step 3.1.1.2

Rewrite the expression.

$x = \sin (45) \frac{11}{\sin (77)}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\sin (45) \frac{11}{\sin (77)}$ .

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Step 3.2.1.1

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = \frac{\sqrt{2}}{2} \cdot \frac{11}{\sin (77)}$

Step 3.2.1.2

Separate fractions.

$x = \frac{\sqrt{2}}{2} (\frac{11}{1} \cdot \frac{1}{\sin (77)})$

Step 3.2.1.3

Convert from $\frac{1}{\sin (77)}$ to $\csc (77)$ .

$x = \frac{\sqrt{2}}{2} (\frac{11}{1} \csc (77))$

Step 3.2.1.4

Divide $11$ by $1$ .

$x = \frac{\sqrt{2}}{2} (11 \csc (77))$

Step 3.2.1.5

Evaluate $\csc (77)$ .

$x = \frac{\sqrt{2}}{2} (11 \cdot 1.0263041)$

Step 3.2.1.6

Multiply $11$ by $1.0263041$ .

$x = \frac{\sqrt{2}}{2} \cdot 11.28934518$

Step 3.2.1.7

Multiply $\frac{\sqrt{2}}{2} \cdot 11.28934518$ .

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Step 3.2.1.7.1

Combine $\frac{\sqrt{2}}{2}$ and $11.28934518$ .

$x = \frac{\sqrt{2} \cdot 11.28934518}{2}$

Step 3.2.1.7.2

Multiply $\sqrt{2}$ by $11.28934518$ .

$x = \frac{15.96554507}{2}$

Step 3.2.1.8

Divide $15.96554507$ by $2$ .

$x = 7.98277253$