Trigonometry Examples

$\frac{200 \sin (105)}{\sin (22)} = x$

Step 1

Rewrite the equation as $x = \frac{200 \sin (105)}{\sin (22)}$ .

$x = \frac{200 \sin (105)}{\sin (22)}$

Step 2

Simplify $\frac{200 \sin (105)}{\sin (22)}$ .

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Step 2.1

Separate fractions.

$x = \frac{200}{1} \cdot \frac{\sin (105)}{\sin (22)}$

Step 2.2

Rewrite $\frac{\sin (105)}{\sin (22)}$ as a product.

$x = \frac{200}{1} (\sin (105) \frac{1}{\sin (22)})$

Step 2.3

Write $\sin (105)$ as a fraction with denominator $1$ .

$x = \frac{200}{1} (\frac{\sin (105)}{1} \cdot \frac{1}{\sin (22)})$

Step 2.4

Simplify.

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Step 2.4.1

Divide $\sin (105)$ by $1$ .

$x = \frac{200}{1} (\sin (105) \frac{1}{\sin (22)})$

Step 2.4.2

Convert from $\frac{1}{\sin (22)}$ to $\csc (22)$ .

$x = \frac{200}{1} (\sin (105) \csc (22))$

Step 2.5

Divide $200$ by $1$ .

$x = 200 (\sin (105) \csc (22))$

Step 2.6

The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 2.6.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$x = 200 (\sin (75) \csc (22))$

Step 2.6.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$x = 200 (\sin (30 + 45) \csc (22))$

Step 2.6.3

Apply the sum of angles identity.

$x = 200 ((\sin (30) \cos (45) + \cos (30) \sin (45)) \csc (22))$

Step 2.6.4

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

$x = 200 ((\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \csc (22))$

Step 2.6.5

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

$x = 200 ((\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \csc (22))$

Step 2.6.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$x = 200 ((\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) \csc (22))$

Step 2.6.7

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

$x = 200 ((\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \csc (22))$

Step 2.6.8

Simplify $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 2.6.8.1

Simplify each term.

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Step 2.6.8.1.1

Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 2.6.8.1.1.1

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

$x = 200 ((\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \csc (22))$

Step 2.6.8.1.1.2

Multiply $2$ by $2$ .

$x = 200 ((\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \csc (22))$

Step 2.6.8.1.2

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

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Step 2.6.8.1.2.1

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

$x = 200 ((\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \csc (22))$

Step 2.6.8.1.2.2

Combine using the product rule for radicals.

$x = 200 ((\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \csc (22))$

Step 2.6.8.1.2.3

Multiply $3$ by $2$ .

$x = 200 ((\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \csc (22))$

Step 2.6.8.1.2.4

Multiply $2$ by $2$ .

$x = 200 ((\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) \csc (22))$

Step 2.6.8.2

Combine the numerators over the common denominator.

$x = 200 (\frac{\sqrt{2} + \sqrt{6}}{4} \csc (22))$

Step 2.7

Evaluate $\csc (22)$ .

$x = 200 (\frac{\sqrt{2} + \sqrt{6}}{4} \cdot 2.66946716)$

Step 2.8

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot 2.66946716$ .

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Step 2.8.1

Combine $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $2.66946716$ .

$x = 200 \frac{(\sqrt{2} + \sqrt{6}) \cdot 2.66946716}{4}$

Step 2.8.2

Multiply $\sqrt{2} + \sqrt{6}$ by $2.66946716$ .

$x = 200 (\frac{10.31402909}{4})$

Step 2.9

Cancel the common factor of $4$ .

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Step 2.9.1

Factor $4$ out of $200$ .

$x = 4 (50) \frac{10.31402909}{4}$

Step 2.9.2

Cancel the common factor.

$x = 4 \cdot 50 \frac{10.31402909}{4}$

Step 2.9.3

Rewrite the expression.

$x = 50 \cdot 10.31402909$

Step 2.10

Multiply $50$ by $10.31402909$ .

$x = 515.70145494$