Trigonometry Examples

2 sin (\frac{π}{3} x) = \sqrt{2}

$2 \sin (\frac{π}{3} x) = \sqrt{2}$

Step 1

Divide each term in

2 sin (\frac{π}{3} x) = \sqrt{2}

$2 \sin (\frac{π}{3} x) = \sqrt{2}$ by

2

$2$ and simplify.

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Step 1.1

Divide each term in

2 sin (\frac{π}{3} x) = \sqrt{2}

$2 \sin (\frac{π}{3} x) = \sqrt{2}$ by

2

$2$ .

\frac{2 sin (\frac{π}{3} x)}{2} = \frac{\sqrt{2}}{2}

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

2

$2$ .

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Step 1.2.1.1

Cancel the common factor.

\frac{2 sin (\frac{π}{3} x)}{2} = \frac{\sqrt{2}}{2}

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

Step 1.2.1.2

Divide

sin (\frac{π}{3} x)

$\sin (\frac{π}{3} x)$ by

1

$1$ .

sin (\frac{π}{3} x) = \frac{\sqrt{2}}{2}

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

sin (\frac{π}{3} x) = \frac{\sqrt{2}}{2}

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

sin (\frac{π}{3} x) = \frac{\sqrt{2}}{2}

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

sin (\frac{π}{3} x) = \frac{\sqrt{2}}{2}

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

Step 2

Take the inverse sine of both sides of the equation to extract

x

$x$ from inside the sine.

\frac{π}{3} x = arcsin (\frac{\sqrt{2}}{2})

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

Step 3

Simplify the left side.

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Step 3.1

Combine

\frac{π}{3}

$\frac{π}{3}$ and

x

$x$ .

\frac{π x}{3} = arcsin (\frac{\sqrt{2}}{2})

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

\frac{π x}{3} = arcsin (\frac{\sqrt{2}}{2})

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of

arcsin (\frac{\sqrt{2}}{2})

$\arcsin (\frac{\sqrt{2}}{2})$ is

\frac{π}{4}

$\frac{π}{4}$ .

\frac{π x}{3} = \frac{π}{4}

$\frac{π x}{3} = \frac{π}{4}$

\frac{π x}{3} = \frac{π}{4}

$\frac{π x}{3} = \frac{π}{4}$

Step 5

Multiply both sides of the equation by

\frac{3}{π}

$\frac{3}{π}$ .

\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} \cdot \frac{π}{4}

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} \cdot \frac{π}{4}$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify

\frac{3}{π} \cdot \frac{π x}{3}

$\frac{3}{π} \cdot \frac{π x}{3}$ .

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Step 6.1.1.1

Cancel the common factor of

3

$3$ .

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Step 6.1.1.1.1

Cancel the common factor.

\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} \cdot \frac{π}{4}

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} \cdot \frac{π}{4}$

Step 6.1.1.1.2

Rewrite the expression.

\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{π}{4}

$\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{π}{4}$

\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{π}{4}

$\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{π}{4}$

Step 6.1.1.2

Cancel the common factor of

π

$π$ .

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Step 6.1.1.2.1

Factor

π

$π$ out of

π x

$π x$ .

\frac{1}{π} (π (x)) = \frac{3}{π} \cdot \frac{π}{4}

$\frac{1}{π} (π (x)) = \frac{3}{π} \cdot \frac{π}{4}$

Step 6.1.1.2.2

Cancel the common factor.

\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{π}{4}

$\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{π}{4}$

Step 6.1.1.2.3

Rewrite the expression.

x = \frac{3}{π} \cdot \frac{π}{4}

$x = \frac{3}{π} \cdot \frac{π}{4}$

x = \frac{3}{π} \cdot \frac{π}{4}

$x = \frac{3}{π} \cdot \frac{π}{4}$

x = \frac{3}{π} \cdot \frac{π}{4}

$x = \frac{3}{π} \cdot \frac{π}{4}$

x = \frac{3}{π} \cdot \frac{π}{4}

$x = \frac{3}{π} \cdot \frac{π}{4}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify

\frac{3}{π} \cdot \frac{π}{4}

$\frac{3}{π} \cdot \frac{π}{4}$ .

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Step 6.2.1.1

Cancel the common factor of

π

$π$ .

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Step 6.2.1.1.1

Cancel the common factor.

x = \frac{3}{π} \cdot \frac{π}{4}

$x = \frac{3}{π} \cdot \frac{π}{4}$

Step 6.2.1.1.2

Rewrite the expression.

x = 3 (\frac{1}{4})

$x = 3 (\frac{1}{4})$

x = 3 (\frac{1}{4})

$x = 3 (\frac{1}{4})$

Step 6.2.1.2

Combine

3

$3$ and

\frac{1}{4}

$\frac{1}{4}$ .

x = \frac{3}{4}

$x = \frac{3}{4}$

x = \frac{3}{4}

$x = \frac{3}{4}$

x = \frac{3}{4}

$x = \frac{3}{4}$

x = \frac{3}{4}

$x = \frac{3}{4}$

Step 7

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

π

$π$ to find the solution in the second quadrant.

\frac{π x}{3} = π - \frac{π}{4}

$\frac{π x}{3} = π - \frac{π}{4}$

Step 8

Solve for

x

$x$ .

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Step 8.1

Multiply both sides of the equation by

\frac{3}{π}

$\frac{3}{π}$ .

\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π - \frac{π}{4})

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π - \frac{π}{4})$

Step 8.2

Simplify both sides of the equation.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Simplify

\frac{3}{π} \cdot \frac{π x}{3}

$\frac{3}{π} \cdot \frac{π x}{3}$ .

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Step 8.2.1.1.1

Cancel the common factor of

3

$3$ .

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Step 8.2.1.1.1.1

Cancel the common factor.

\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π - \frac{π}{4})

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π - \frac{π}{4})$

Step 8.2.1.1.1.2

Rewrite the expression.

\frac{1}{π} (π x) = \frac{3}{π} (π - \frac{π}{4})

$\frac{1}{π} (π x) = \frac{3}{π} (π - \frac{π}{4})$

\frac{1}{π} (π x) = \frac{3}{π} (π - \frac{π}{4})

$\frac{1}{π} (π x) = \frac{3}{π} (π - \frac{π}{4})$

Step 8.2.1.1.2

Cancel the common factor of

π

$π$ .

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Step 8.2.1.1.2.1

Factor

π

$π$ out of

π x

$π x$ .

\frac{1}{π} (π (x)) = \frac{3}{π} (π - \frac{π}{4})

$\frac{1}{π} (π (x)) = \frac{3}{π} (π - \frac{π}{4})$

Step 8.2.1.1.2.2

Cancel the common factor.

\frac{1}{π} (π x) = \frac{3}{π} (π - \frac{π}{4})

$\frac{1}{π} (π x) = \frac{3}{π} (π - \frac{π}{4})$

Step 8.2.1.1.2.3

Rewrite the expression.

x = \frac{3}{π} (π - \frac{π}{4})

$x = \frac{3}{π} (π - \frac{π}{4})$

x = \frac{3}{π} (π - \frac{π}{4})

$x = \frac{3}{π} (π - \frac{π}{4})$

x = \frac{3}{π} (π - \frac{π}{4})

$x = \frac{3}{π} (π - \frac{π}{4})$

x = \frac{3}{π} (π - \frac{π}{4})

$x = \frac{3}{π} (π - \frac{π}{4})$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Simplify

\frac{3}{π} (π - \frac{π}{4})

$\frac{3}{π} (π - \frac{π}{4})$ .

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Step 8.2.2.1.1

To write

π

$π$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

x = \frac{3}{π} (π \cdot \frac{4}{4} - \frac{π}{4})

$x = \frac{3}{π} (π \cdot \frac{4}{4} - \frac{π}{4})$

Step 8.2.2.1.2

Combine fractions.

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Step 8.2.2.1.2.1

Combine

π

$π$ and

\frac{4}{4}

$\frac{4}{4}$ .

x = \frac{3}{π} (\frac{π \cdot 4}{4} - \frac{π}{4})

$x = \frac{3}{π} (\frac{π \cdot 4}{4} - \frac{π}{4})$

Step 8.2.2.1.2.2

Combine the numerators over the common denominator.

x = \frac{3}{π} \cdot \frac{π \cdot 4 - π}{4}

$x = \frac{3}{π} \cdot \frac{π \cdot 4 - π}{4}$

x = \frac{3}{π} \cdot \frac{π \cdot 4 - π}{4}

$x = \frac{3}{π} \cdot \frac{π \cdot 4 - π}{4}$

Step 8.2.2.1.3

Simplify the numerator.

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Step 8.2.2.1.3.1

Move

4

$4$ to the left of

π

$π$ .

x = \frac{3}{π} \cdot \frac{4 \cdot π - π}{4}

$x = \frac{3}{π} \cdot \frac{4 \cdot π - π}{4}$

Step 8.2.2.1.3.2

Subtract

π

$π$ from

4 π

$4 π$ .

x = \frac{3}{π} \cdot \frac{3 π}{4}

$x = \frac{3}{π} \cdot \frac{3 π}{4}$

x = \frac{3}{π} \cdot \frac{3 π}{4}

$x = \frac{3}{π} \cdot \frac{3 π}{4}$

Step 8.2.2.1.4

Simplify terms.

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Step 8.2.2.1.4.1

Cancel the common factor of

π

$π$ .

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Step 8.2.2.1.4.1.1

Factor

π

$π$ out of

3 π

$3 π$ .

x = \frac{3}{π} \cdot \frac{π \cdot 3}{4}

$x = \frac{3}{π} \cdot \frac{π \cdot 3}{4}$

Step 8.2.2.1.4.1.2

Cancel the common factor.

x = \frac{3}{π} \cdot \frac{π \cdot 3}{4}

$x = \frac{3}{π} \cdot \frac{π \cdot 3}{4}$

Step 8.2.2.1.4.1.3

Rewrite the expression.

x = 3 (\frac{3}{4})

$x = 3 (\frac{3}{4})$

x = 3 (\frac{3}{4})

$x = 3 (\frac{3}{4})$

Step 8.2.2.1.4.2

Combine

3

$3$ and

\frac{3}{4}

$\frac{3}{4}$ .

x = \frac{3 \cdot 3}{4}

$x = \frac{3 \cdot 3}{4}$

Step 8.2.2.1.4.3

Multiply

3

$3$ by

3

$3$ .

x = \frac{9}{4}

$x = \frac{9}{4}$

x = \frac{9}{4}

$x = \frac{9}{4}$

x = \frac{9}{4}

$x = \frac{9}{4}$

x = \frac{9}{4}

$x = \frac{9}{4}$

x = \frac{9}{4}

$x = \frac{9}{4}$

x = \frac{9}{4}

$x = \frac{9}{4}$

Step 9

Find the period of

sin (\frac{π}{3} x)

$\sin (\frac{π}{3} x)$ .

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Step 9.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 9.2

Replace

$b$ with

$\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

Step 9.3

$\frac{π}{3}$ is approximately

$1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 9.5

Cancel the common factor of

$π$ .

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Step 9.5.1

Factor

$π$ out of

$2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 9.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 9.5.3

Rewrite the expression.

$2 \cdot 3$

Step 9.6

Multiply

$2$ by

$3$ .

$6$

Step 10

The period of the

$\sin (\frac{π}{3} x)$ function is

$6$ so values will repeat every

$6$ radians in both directions.

$x = \frac{3}{4} + 6 n, \frac{9}{4} + 6 n$ , for any integer

$n$