Trigonometry Examples

$8 \sin (6 x) = 7$

Step 1

Divide each term in $8 \sin (6 x) = 7$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 \sin (6 x) = 7$ by $8$ .

$\frac{8 \sin (6 x)}{8} = \frac{7}{8}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 \sin (6 x)}{8} = \frac{7}{8}$

Step 1.2.1.2

Divide $\sin (6 x)$ by $1$ .

$\sin (6 x) = \frac{7}{8}$

Step 2

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

$6 x = \arcsin (\frac{7}{8})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (\frac{7}{8})$ .

$6 x = 1.06543581$

Step 4

Divide each term in $6 x = 1.06543581$ by $6$ and simplify.

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Step 4.1

Divide each term in $6 x = 1.06543581$ by $6$ .

$\frac{6 x}{6} = \frac{1.06543581}{6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $6$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{1.06543581}{6}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.06543581}{6}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $1.06543581$ by $6$ .

$x = 0.17757263$

Step 5

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

$6 x = (3.14159265) - 1.06543581$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $1.06543581$ from $3.14159265$ .

$6 x = 2.07615683$

Step 6.2

Divide each term in $6 x = 2.07615683$ by $6$ and simplify.

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Step 6.2.1

Divide each term in $6 x = 2.07615683$ by $6$ .

$\frac{6 x}{6} = \frac{2.07615683}{6}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $6$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{2.07615683}{6}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.07615683}{6}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $2.07615683$ by $6$ .

$x = 0.34602613$

Step 7

Find the period of $\sin (6 x)$ .

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Step 7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 7.4

Cancel the common factor of $2$ and $6$ .

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Step 7.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 7.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 7.4.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 8

The period of the $\sin (6 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = 0.17757263 + \frac{π n}{3}, 0.34602613 + \frac{π n}{3}$ , for any integer $n$