Trigonometry Examples

$\frac{3 π}{4} + 2 n π = 12 \sin (x) \cos (x) - 3 \sqrt{2}$

Step 1

Rewrite the equation as $12 \sin (x) \cos (x) - 3 \sqrt{2} = \frac{3 π}{4} + 2 n π$ .

$12 \sin (x) \cos (x) - 3 \sqrt{2} = \frac{3 π}{4} + 2 n π$

Step 2

Add $3 \sqrt{2}$ to both sides of the equation.

$12 \sin (x) \cos (x) = \frac{3 π}{4} + 2 n π + 3 \sqrt{2}$

Step 3

Multiply each term in $12 \sin (x) \cos (x) = \frac{3 π}{4} + 2 n π + 3 \sqrt{2}$ by $\frac{1}{6}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $12 \sin (x) \cos (x) = \frac{3 π}{4} + 2 n π + 3 \sqrt{2}$ by $\frac{1}{6}$ .

$12 \sin (x) \cos (x) \frac{1}{6} = \frac{3 π}{4} \cdot \frac{1}{6} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $6$ .

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Step 3.2.1.1

Factor $6$ out of $12 \sin (x) \cos (x)$ .

$6 (2 \sin (x) \cos (x)) \frac{1}{6} = \frac{3 π}{4} \cdot \frac{1}{6} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.2.1.2

Cancel the common factor.

$6 (2 \sin (x) \cos (x)) \frac{1}{6} = \frac{3 π}{4} \cdot \frac{1}{6} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.2.1.3

Rewrite the expression.

$2 \sin (x) \cos (x) = \frac{3 π}{4} \cdot \frac{1}{6} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.2.2

Apply the sine double-angle identity.

$\sin (2 x) = \frac{3 π}{4} \cdot \frac{1}{6} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Factor $3$ out of $3 π$ .

$\sin (2 x) = \frac{3 (π)}{4} \cdot \frac{1}{6} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.1.2

Factor $3$ out of $6$ .

$\sin (2 x) = \frac{3 (π)}{4} \cdot \frac{1}{3 (2)} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.1.3

Cancel the common factor.

$\sin (2 x) = \frac{3 π}{4} \cdot \frac{1}{3 \cdot 2} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.1.4

Rewrite the expression.

$\sin (2 x) = \frac{π}{4} \cdot \frac{1}{2} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.2

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

$\sin (2 x) = \frac{π}{4 \cdot 2} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.3

Multiply $4$ by $2$ .

$\sin (2 x) = \frac{π}{8} + 2 n π \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Factor $2$ out of $2 n π$ .

$\sin (2 x) = \frac{π}{8} + 2 (n π) \frac{1}{6} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.4.2

Factor $2$ out of $6$ .

$\sin (2 x) = \frac{π}{8} + 2 (n π) \frac{1}{2 (3)} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.4.3

Cancel the common factor.

$\sin (2 x) = \frac{π}{8} + 2 (n π) \frac{1}{2 \cdot 3} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.4.4

Rewrite the expression.

$\sin (2 x) = \frac{π}{8} + n π \frac{1}{3} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.5

Combine $\frac{1}{3}$ and $n$ .

$\sin (2 x) = \frac{π}{8} + \frac{n}{3} π + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.6

Combine $\frac{n}{3}$ and $π$ .

$\sin (2 x) = \frac{π}{8} + \frac{n π}{3} + 3 \sqrt{2} \frac{1}{6}$

Step 3.3.1.7

Cancel the common factor of $3$ .

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Step 3.3.1.7.1

Factor $3$ out of $3 \sqrt{2}$ .

$\sin (2 x) = \frac{π}{8} + \frac{n π}{3} + 3 (\sqrt{2}) \frac{1}{6}$

Step 3.3.1.7.2

Factor $3$ out of $6$ .

$\sin (2 x) = \frac{π}{8} + \frac{n π}{3} + 3 (\sqrt{2}) \frac{1}{3 (2)}$

Step 3.3.1.7.3

Cancel the common factor.

$\sin (2 x) = \frac{π}{8} + \frac{n π}{3} + 3 \sqrt{2} \frac{1}{3 \cdot 2}$

Step 3.3.1.7.4

Rewrite the expression.

$\sin (2 x) = \frac{π}{8} + \frac{n π}{3} + \sqrt{2} \frac{1}{2}$

Step 3.3.1.8

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

$\sin (2 x) = \frac{π}{8} + \frac{n π}{3} + \frac{\sqrt{2}}{2}$

Step 4

Simplify $\frac{π}{8} + \frac{n π}{3} + \frac{\sqrt{2}}{2}$ .

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Step 4.1

Reorder $n$ and $π$ .

$\sin (2 x) = \frac{π}{8} + \frac{π n}{3} + \frac{\sqrt{2}}{2}$

Step 4.2

Reorder $\frac{π}{8}$ and $\frac{π n}{3}$ .

$\sin (2 x) = \frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2}$

Step 5

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

$2 x = \arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2})$

Step 6

Divide each term in $2 x = \arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2})$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = \arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2})$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2})}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2})}{2}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2})}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify the numerator.

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Step 6.3.1.1

To write $\frac{\sqrt{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{\arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2}}{2} \cdot \frac{4}{4})}{2}$

Step 6.3.1.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{4}{4}$ .

$x = \frac{\arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2} \cdot 4}{2 \cdot 4})}{2}$

Step 6.3.1.2.2

Multiply $2$ by $4$ .

$x = \frac{\arcsin (\frac{π n}{3} + \frac{π}{8} + \frac{\sqrt{2} \cdot 4}{8})}{2}$

Step 6.3.1.3

Combine the numerators over the common denominator.

$x = \frac{\arcsin (\frac{π n}{3} + \frac{π + \sqrt{2} \cdot 4}{8})}{2}$

Step 6.3.1.4

Move $4$ to the left of $\sqrt{2}$ .

$x = \frac{\arcsin (\frac{π n}{3} + \frac{π + 4 \sqrt{2}}{8})}{2}$