Trigonometry Examples

$\sin (75) = \frac{x}{6}$

Step 1

Rewrite the equation as $\frac{x}{6} = \sin (75)$ .

$\frac{x}{6} = \sin (75)$

Step 2

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 \sin (75)$

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 $6$ .

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Step 3.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 \sin (75)$

Step 3.1.1.2

Rewrite the expression.

$x = 6 \sin (75)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $6 \sin (75)$ .

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Step 3.2.1.1

The exact value of $\sin (75)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 3.2.1.1.1

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

$x = 6 \sin (30 + 45)$

Step 3.2.1.1.2

Apply the sum of angles identity.

$x = 6 (\sin (30) \cos (45) + \cos (30) \sin (45))$

Step 3.2.1.1.3

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

$x = 6 (\frac{1}{2} \cos (45) + \cos (30) \sin (45))$

Step 3.2.1.1.4

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

$x = 6 (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45))$

Step 3.2.1.1.5

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

$x = 6 (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45))$

Step 3.2.1.1.6

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

$x = 6 (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$

Step 3.2.1.1.7

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

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Step 3.2.1.1.7.1

Simplify each term.

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Step 3.2.1.1.7.1.1

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

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Step 3.2.1.1.7.1.1.1

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

$x = 6 (\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$

Step 3.2.1.1.7.1.1.2

Multiply $2$ by $2$ .

$x = 6 (\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$

Step 3.2.1.1.7.1.2

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

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Step 3.2.1.1.7.1.2.1

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

$x = 6 (\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2})$

Step 3.2.1.1.7.1.2.2

Combine using the product rule for radicals.

$x = 6 (\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2})$

Step 3.2.1.1.7.1.2.3

Multiply $3$ by $2$ .

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

Step 3.2.1.1.7.1.2.4

Multiply $2$ by $2$ .

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

Step 3.2.1.1.7.2

Combine the numerators over the common denominator.

$x = 6 \frac{\sqrt{2} + \sqrt{6}}{4}$

Step 3.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.2.1

Factor $2$ out of $6$ .

$x = 2 (3) \frac{\sqrt{2} + \sqrt{6}}{4}$

Step 3.2.1.2.2

Factor $2$ out of $4$ .

$x = 2 \cdot 3 \frac{\sqrt{2} + \sqrt{6}}{2 \cdot 2}$

Step 3.2.1.2.3

Cancel the common factor.

$x = 2 \cdot 3 \frac{\sqrt{2} + \sqrt{6}}{2 \cdot 2}$

Step 3.2.1.2.4

Rewrite the expression.

$x = 3 \frac{\sqrt{2} + \sqrt{6}}{2}$

Step 3.2.1.3

Combine $3$ and $\frac{\sqrt{2} + \sqrt{6}}{2}$ .

$x = \frac{3 (\sqrt{2} + \sqrt{6})}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 (\sqrt{2} + \sqrt{6})}{2}$

Decimal Form:

$x = 5.79555495 \dots$