Trigonometry Examples

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

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

Step 1

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

x

$x$ from inside the sine.

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

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

Step 2

Simplify the right side.

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Step 2.1

The exact value of

\arcsin (- \frac{\sqrt{3}}{2})

$\arcsin (- \frac{\sqrt{3}}{2})$ is

- \frac{π}{3}

$- \frac{π}{3}$ .

3 x = - \frac{π}{3}

$3 x = - \frac{π}{3}$

3 x = - \frac{π}{3}

$3 x = - \frac{π}{3}$

Step 3

Divide each term in

3 x = - \frac{π}{3}

$3 x = - \frac{π}{3}$ by

3

$3$ and simplify.

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Step 3.1

Divide each term in

3 x = - \frac{π}{3}

$3 x = - \frac{π}{3}$ by

3

$3$ .

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

3

$3$ .

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Step 3.2.1.1

Cancel the common factor.

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

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

Step 3.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- \frac{π}{3}}{3}

$x = \frac{- \frac{π}{3}}{3}$

x = \frac{- \frac{π}{3}}{3}

$x = \frac{- \frac{π}{3}}{3}$

x = \frac{- \frac{π}{3}}{3}

$x = \frac{- \frac{π}{3}}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

x = - \frac{π}{3} \cdot \frac{1}{3}

$x = - \frac{π}{3} \cdot \frac{1}{3}$

Step 3.3.2

Multiply

- \frac{π}{3} \cdot \frac{1}{3}

$- \frac{π}{3} \cdot \frac{1}{3}$ .

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Step 3.3.2.1

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

\frac{π}{3}

$\frac{π}{3}$ .

x = - \frac{π}{3 \cdot 3}

$x = - \frac{π}{3 \cdot 3}$

Step 3.3.2.2

Multiply

3

$3$ by

3

$3$ .

x = - \frac{π}{9}

$x = - \frac{π}{9}$

x = - \frac{π}{9}

$x = - \frac{π}{9}$

x = - \frac{π}{9}

$x = - \frac{π}{9}$

x = - \frac{π}{9}

$x = - \frac{π}{9}$

Step 4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

2 π

$2 π$ , to find a reference angle. Next, add this reference angle to

π

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

3 x = 2 π + \frac{π}{3} + π

$3 x = 2 π + \frac{π}{3} + π$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract

2 π

$2 π$ from

2 π + \frac{π}{3} + π

$2 π + \frac{π}{3} + π$ .

3 x = 2 π + \frac{π}{3} + π - 2 π

$3 x = 2 π + \frac{π}{3} + π - 2 π$

Step 5.2

The resulting angle of

\frac{4 π}{3}

$\frac{4 π}{3}$ is positive, less than

2 π

$2 π$ , and coterminal with

2 π + \frac{π}{3} + π

$2 π + \frac{π}{3} + π$ .

3 x = \frac{4 π}{3}

$3 x = \frac{4 π}{3}$

Step 5.3

Divide each term in

3 x = \frac{4 π}{3}

$3 x = \frac{4 π}{3}$ by

3

$3$ and simplify.

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Step 5.3.1

Divide each term in

3 x = \frac{4 π}{3}

$3 x = \frac{4 π}{3}$ by

3

$3$ .

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of

3

$3$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

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

Step 5.3.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\frac{4 π}{3}}{3}

$x = \frac{\frac{4 π}{3}}{3}$

x = \frac{\frac{4 π}{3}}{3}

$x = \frac{\frac{4 π}{3}}{3}$

x = \frac{\frac{4 π}{3}}{3}

$x = \frac{\frac{4 π}{3}}{3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

x = \frac{4 π}{3} \cdot \frac{1}{3}

$x = \frac{4 π}{3} \cdot \frac{1}{3}$

Step 5.3.3.2

Multiply

\frac{4 π}{3} \cdot \frac{1}{3}

$\frac{4 π}{3} \cdot \frac{1}{3}$ .

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Step 5.3.3.2.1

Multiply

\frac{4 π}{3}

$\frac{4 π}{3}$ by

\frac{1}{3}

$\frac{1}{3}$ .

x = \frac{4 π}{3 \cdot 3}

$x = \frac{4 π}{3 \cdot 3}$

Step 5.3.3.2.2

Multiply

3

$3$ by

3

$3$ .

x = \frac{4 π}{9}

$x = \frac{4 π}{9}$

x = \frac{4 π}{9}

$x = \frac{4 π}{9}$

x = \frac{4 π}{9}

$x = \frac{4 π}{9}$

x = \frac{4 π}{9}

$x = \frac{4 π}{9}$

x = \frac{4 π}{9}

$x = \frac{4 π}{9}$

Step 6

Find the period of

\sin (3 x)

$\sin (3 x)$ .

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Step 6.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 6.2

Replace

b

$b$ with

3

$3$ in the formula for period.

\frac{2 π}{| 3 |}

$\frac{2 π}{| 3 |}$

Step 6.3

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

0

$0$ and

3

$3$ is

3

$3$ .

\frac{2 π}{3}

$\frac{2 π}{3}$

\frac{2 π}{3}

$\frac{2 π}{3}$

Step 7

Add

\frac{2 π}{3}

$\frac{2 π}{3}$ to every negative angle to get positive angles.

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Step 7.1

Add

\frac{2 π}{3}

$\frac{2 π}{3}$ to

- \frac{π}{9}

$- \frac{π}{9}$ to find the positive angle.

- \frac{π}{9} + \frac{2 π}{3}

$- \frac{π}{9} + \frac{2 π}{3}$

Step 7.2

To write

\frac{2 π}{3}

$\frac{2 π}{3}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{2 π}{3} \cdot \frac{3}{3} - \frac{π}{9}

$\frac{2 π}{3} \cdot \frac{3}{3} - \frac{π}{9}$

Step 7.3

Write each expression with a common denominator of

9

$9$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 7.3.1

Multiply

\frac{2 π}{3}

$\frac{2 π}{3}$ by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{2 π \cdot 3}{3 \cdot 3} - \frac{π}{9}

$\frac{2 π \cdot 3}{3 \cdot 3} - \frac{π}{9}$

Step 7.3.2

Multiply

3

$3$ by

3

$3$ .

\frac{2 π \cdot 3}{9} - \frac{π}{9}

$\frac{2 π \cdot 3}{9} - \frac{π}{9}$

\frac{2 π \cdot 3}{9} - \frac{π}{9}

$\frac{2 π \cdot 3}{9} - \frac{π}{9}$

Step 7.4

Combine the numerators over the common denominator.

\frac{2 π \cdot 3 - π}{9}

$\frac{2 π \cdot 3 - π}{9}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Multiply

3

$3$ by

2

$2$ .

\frac{6 π - π}{9}

$\frac{6 π - π}{9}$

Step 7.5.2

Subtract

π

$π$ from

6 π

$6 π$ .

\frac{5 π}{9}

$\frac{5 π}{9}$

\frac{5 π}{9}

$\frac{5 π}{9}$

Step 7.6

List the new angles.

x = \frac{5 π}{9}

$x = \frac{5 π}{9}$

x = \frac{5 π}{9}

$x = \frac{5 π}{9}$

Step 8

The period of the

\sin (3 x)

$\sin (3 x)$ function is

\frac{2 π}{3}

$\frac{2 π}{3}$ so values will repeat every

\frac{2 π}{3}

$\frac{2 π}{3}$ radians in both directions.

x = \frac{4 π}{9} + \frac{2 π n}{3}, \frac{5 π}{9} + \frac{2 π n}{3}

$x = \frac{4 π}{9} + \frac{2 π n}{3}, \frac{5 π}{9} + \frac{2 π n}{3}$ , for any integer

n

$n$