Trigonometry Examples

$8 \sin (\frac{2}{9} x + \frac{2}{3})$

Step 1

Use the form $a \sin (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 8$

$b = \frac{2}{9}$

$c = - \frac{2}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $8$

Step 3

Find the period of $8 \sin (\frac{2 x}{9} + \frac{2}{3})$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{2}{9}$ in the formula for period.

$\frac{2 π}{| \frac{2}{9} |}$

Step 3.3

$\frac{2}{9}$ is approximately $0.\overline{2}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{9}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{9}{2}$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{9}{2}$

Step 3.5.2

Cancel the common factor.

$2 π \frac{9}{2}$

Step 3.5.3

Rewrite the expression.

$π \cdot 9$

Step 3.6

Move $9$ to the left of $π$ .

$9 π$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- \frac{2}{3}}{\frac{2}{9}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{2}{3} \cdot \frac{9}{2}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

Phase Shift: $\frac{- 2}{3} \cdot \frac{9}{2}$

Step 4.4.2

Factor $2$ out of $- 2$ .

Phase Shift: $\frac{2 (- 1)}{3} \cdot \frac{9}{2}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{2 \cdot - 1}{3} \cdot \frac{9}{2}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- 1}{3} \cdot 9$

Step 4.5

Cancel the common factor of $3$ .

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Step 4.5.1

Factor $3$ out of $9$ .

Phase Shift: $\frac{- 1}{3} \cdot (3 (3))$

Step 4.5.2

Cancel the common factor.

Phase Shift: $\frac{- 1}{3} \cdot (3 \cdot 3)$

Step 4.5.3

Rewrite the expression.

Phase Shift: $- 1 \cdot 3$

Step 4.6

Multiply $- 1$ by $3$ .

Phase Shift: $- 3$

Step 5

List the properties of the trigonometric function.

Amplitude: $8$

Period: $9 π$

Phase Shift: $- 3$ ( $3$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - 3$ .

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Step 6.1.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = 8 \sin (\frac{2 (- 3)}{9} + \frac{2}{3})$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $2$ by $- 3$ .

$f (- 3) = 8 \sin (\frac{- 6}{9} + \frac{2}{3})$

Step 6.1.2.2

Simplify each term.

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Step 6.1.2.2.1

Cancel the common factor of $- 6$ and $9$ .

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Step 6.1.2.2.1.1

Factor $3$ out of $- 6$ .

$f (- 3) = 8 \sin (\frac{3 (- 2)}{9} + \frac{2}{3})$

Step 6.1.2.2.1.2

Cancel the common factors.

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Step 6.1.2.2.1.2.1

Factor $3$ out of $9$ .

$f (- 3) = 8 \sin (\frac{3 \cdot - 2}{3 \cdot 3} + \frac{2}{3})$

Step 6.1.2.2.1.2.2

Cancel the common factor.

$f (- 3) = 8 \sin (\frac{3 \cdot - 2}{3 \cdot 3} + \frac{2}{3})$

Step 6.1.2.2.1.2.3

Rewrite the expression.

$f (- 3) = 8 \sin (\frac{- 2}{3} + \frac{2}{3})$

Step 6.1.2.2.2

Move the negative in front of the fraction.

$f (- 3) = 8 \sin (- \frac{2}{3} + \frac{2}{3})$

Step 6.1.2.3

Combine fractions.

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Step 6.1.2.3.1

Combine the numerators over the common denominator.

$f (- 3) = 8 \sin (\frac{- 2 + 2}{3})$

Step 6.1.2.3.2

Simplify the expression.

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Step 6.1.2.3.2.1

Add $- 2$ and $2$ .

$f (- 3) = 8 \sin (\frac{0}{3})$

Step 6.1.2.3.2.2

Divide $0$ by $3$ .

$f (- 3) = 8 \sin (0)$

Step 6.1.2.4

The exact value of $\sin (0)$ is $0$ .

$f (- 3) = 8 \cdot 0$

Step 6.1.2.5

Multiply $8$ by $0$ .

$f (- 3) = 0$

Step 6.1.2.6

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = \frac{9 π}{4} - 3$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{9 π}{4} - 3$ in the expression.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{2 (\frac{9 π}{4} - 3)}{9} + \frac{2}{3})$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Cancel the common factor of $\frac{9 π}{4} - 3$ and $9$ .

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Step 6.2.2.1.1.1

Factor $3$ out of $2 (\frac{9 π}{4} - 3)$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 (2 (\frac{3 π}{4} - 1))}{9} + \frac{2}{3})$

Step 6.2.2.1.1.2

Cancel the common factors.

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Step 6.2.2.1.1.2.1

Factor $3$ out of $9$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 (2 (\frac{3 π}{4} - 1))}{3 (3)} + \frac{2}{3})$

Step 6.2.2.1.1.2.2

Cancel the common factor.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 (2 (\frac{3 π}{4} - 1))}{3 \cdot 3} + \frac{2}{3})$

Step 6.2.2.1.1.2.3

Rewrite the expression.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{2 (\frac{3 π}{4} - 1)}{3} + \frac{2}{3})$

Step 6.2.2.1.2

Simplify the numerator.

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Step 6.2.2.1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{2 (\frac{3 π}{4} - 1 \cdot \frac{4}{4})}{3} + \frac{2}{3})$

Step 6.2.2.1.2.2

Combine $- 1$ and $\frac{4}{4}$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{2 (\frac{3 π}{4} + \frac{- 1 \cdot 4}{4})}{3} + \frac{2}{3})$

Step 6.2.2.1.2.3

Combine the numerators over the common denominator.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{2 (\frac{3 π - 1 \cdot 4}{4})}{3} + \frac{2}{3})$

Step 6.2.2.1.2.4

Multiply $- 1$ by $4$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{2 (\frac{3 π - 4}{4})}{3} + \frac{2}{3})$

Step 6.2.2.1.3

Combine $2$ and $\frac{3 π - 4}{4}$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{\frac{2 (3 π - 4)}{4}}{3} + \frac{2}{3})$

Step 6.2.2.1.4

Reduce the expression $\frac{2 (3 π - 4)}{4}$ by cancelling the common factors.

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Step 6.2.2.1.4.1

Factor $2$ out of $4$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{\frac{2 (3 π - 4)}{2 \cdot 2}}{3} + \frac{2}{3})$

Step 6.2.2.1.4.2

Cancel the common factor.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{\frac{2 (3 π - 4)}{2 \cdot 2}}{3} + \frac{2}{3})$

Step 6.2.2.1.4.3

Rewrite the expression.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{\frac{3 π - 4}{2}}{3} + \frac{2}{3})$

Step 6.2.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4}{2} \cdot \frac{1}{3} + \frac{2}{3})$

Step 6.2.2.1.6

Multiply $\frac{3 π - 4}{2} \cdot \frac{1}{3}$ .

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Step 6.2.2.1.6.1

Multiply $\frac{3 π - 4}{2}$ by $\frac{1}{3}$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4}{2 \cdot 3} + \frac{2}{3})$

Step 6.2.2.1.6.2

Multiply $2$ by $3$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4}{6} + \frac{2}{3})$

Step 6.2.2.2

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

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4}{6} + \frac{2}{3} \cdot \frac{2}{2})$

Step 6.2.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.3.1

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

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4}{6} + \frac{2 \cdot 2}{3 \cdot 2})$

Step 6.2.2.3.2

Multiply $3$ by $2$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4}{6} + \frac{2 \cdot 2}{6})$

Step 6.2.2.4

Combine the numerators over the common denominator.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4 + 2 \cdot 2}{6})$

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Multiply $2$ by $2$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π - 4 + 4}{6})$

Step 6.2.2.5.2

Add $- 4$ and $4$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π + 0}{6})$

Step 6.2.2.5.3

Add $3 π$ and $0$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π}{6})$

Step 6.2.2.6

Cancel the common factor of $3$ and $6$ .

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Step 6.2.2.6.1

Factor $3$ out of $3 π$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 (π)}{6})$

Step 6.2.2.6.2

Cancel the common factors.

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Step 6.2.2.6.2.1

Factor $3$ out of $6$ .

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.6.2.2

Cancel the common factor.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.6.2.3

Rewrite the expression.

$f (\frac{9 π}{4} - 3) = 8 \sin (\frac{π}{2})$

Step 6.2.2.7

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

$f (\frac{9 π}{4} - 3) = 8 \cdot 1$

Step 6.2.2.8

Multiply $8$ by $1$ .

$f (\frac{9 π}{4} - 3) = 8$

Step 6.2.2.9

The final answer is $8$ .

$8$

Step 6.3

Find the point at $x = \frac{9 π}{2} - 3$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{9 π}{2} - 3$ in the expression.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{2 (\frac{9 π}{2} - 3)}{9} + \frac{2}{3})$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Cancel the common factor of $\frac{9 π}{2} - 3$ and $9$ .

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Step 6.3.2.1.1.1

Factor $3$ out of $2 (\frac{9 π}{2} - 3)$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 (2 (\frac{3 π}{2} - 1))}{9} + \frac{2}{3})$

Step 6.3.2.1.1.2

Cancel the common factors.

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Step 6.3.2.1.1.2.1

Factor $3$ out of $9$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 (2 (\frac{3 π}{2} - 1))}{3 (3)} + \frac{2}{3})$

Step 6.3.2.1.1.2.2

Cancel the common factor.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 (2 (\frac{3 π}{2} - 1))}{3 \cdot 3} + \frac{2}{3})$

Step 6.3.2.1.1.2.3

Rewrite the expression.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{2 (\frac{3 π}{2} - 1)}{3} + \frac{2}{3})$

Step 6.3.2.1.2

Simplify the numerator.

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Step 6.3.2.1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{2 (\frac{3 π}{2} - 1 \cdot \frac{2}{2})}{3} + \frac{2}{3})$

Step 6.3.2.1.2.2

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

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{2 (\frac{3 π}{2} + \frac{- 1 \cdot 2}{2})}{3} + \frac{2}{3})$

Step 6.3.2.1.2.3

Combine the numerators over the common denominator.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{2 (\frac{3 π - 1 \cdot 2}{2})}{3} + \frac{2}{3})$

Step 6.3.2.1.2.4

Multiply $- 1$ by $2$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{2 (\frac{3 π - 2}{2})}{3} + \frac{2}{3})$

Step 6.3.2.1.3

Combine $2$ and $\frac{3 π - 2}{2}$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{\frac{2 (3 π - 2)}{2}}{3} + \frac{2}{3})$

Step 6.3.2.1.4

Reduce the expression by cancelling the common factors.

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Step 6.3.2.1.4.1

Reduce the expression $\frac{2 (3 π - 2)}{2}$ by cancelling the common factors.

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Step 6.3.2.1.4.1.1

Cancel the common factor.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{\frac{2 (3 π - 2)}{2}}{3} + \frac{2}{3})$

Step 6.3.2.1.4.1.2

Rewrite the expression.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{\frac{3 π - 2}{1}}{3} + \frac{2}{3})$

Step 6.3.2.1.4.2

Divide $3 π - 2$ by $1$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 π - 2}{3} + \frac{2}{3})$

Step 6.3.2.2

Simplify terms.

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Step 6.3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 π - 2 + 2}{3})$

Step 6.3.2.2.2

Simplify by adding numbers.

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Step 6.3.2.2.2.1

Add $- 2$ and $2$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 π + 0}{3})$

Step 6.3.2.2.2.2

Add $3 π$ and $0$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 π}{3})$

Step 6.3.2.2.3

Cancel the common factor of $3$ .

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Step 6.3.2.2.3.1

Cancel the common factor.

$f (\frac{9 π}{2} - 3) = 8 \sin (\frac{3 π}{3})$

Step 6.3.2.2.3.2

Divide $π$ by $1$ .

$f (\frac{9 π}{2} - 3) = 8 \sin (π)$

Step 6.3.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (\frac{9 π}{2} - 3) = 8 \sin (0)$

Step 6.3.2.4

The exact value of $\sin (0)$ is $0$ .

$f (\frac{9 π}{2} - 3) = 8 \cdot 0$

Step 6.3.2.5

Multiply $8$ by $0$ .

$f (\frac{9 π}{2} - 3) = 0$

Step 6.3.2.6

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{27 π}{4} - 3$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{27 π}{4} - 3$ in the expression.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{2 (\frac{27 π}{4} - 3)}{9} + \frac{2}{3})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Cancel the common factor of $\frac{27 π}{4} - 3$ and $9$ .

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Step 6.4.2.1.1.1

Factor $3$ out of $2 (\frac{27 π}{4} - 3)$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 (2 (\frac{9 π}{4} - 1))}{9} + \frac{2}{3})$

Step 6.4.2.1.1.2

Cancel the common factors.

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Step 6.4.2.1.1.2.1

Factor $3$ out of $9$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 (2 (\frac{9 π}{4} - 1))}{3 (3)} + \frac{2}{3})$

Step 6.4.2.1.1.2.2

Cancel the common factor.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 (2 (\frac{9 π}{4} - 1))}{3 \cdot 3} + \frac{2}{3})$

Step 6.4.2.1.1.2.3

Rewrite the expression.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{2 (\frac{9 π}{4} - 1)}{3} + \frac{2}{3})$

Step 6.4.2.1.2

Simplify the numerator.

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Step 6.4.2.1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{2 (\frac{9 π}{4} - 1 \cdot \frac{4}{4})}{3} + \frac{2}{3})$

Step 6.4.2.1.2.2

Combine $- 1$ and $\frac{4}{4}$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{2 (\frac{9 π}{4} + \frac{- 1 \cdot 4}{4})}{3} + \frac{2}{3})$

Step 6.4.2.1.2.3

Combine the numerators over the common denominator.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{2 (\frac{9 π - 1 \cdot 4}{4})}{3} + \frac{2}{3})$

Step 6.4.2.1.2.4

Multiply $- 1$ by $4$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{2 (\frac{9 π - 4}{4})}{3} + \frac{2}{3})$

Step 6.4.2.1.3

Combine $2$ and $\frac{9 π - 4}{4}$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{\frac{2 (9 π - 4)}{4}}{3} + \frac{2}{3})$

Step 6.4.2.1.4

Reduce the expression $\frac{2 (9 π - 4)}{4}$ by cancelling the common factors.

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Step 6.4.2.1.4.1

Factor $2$ out of $4$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{\frac{2 (9 π - 4)}{2 \cdot 2}}{3} + \frac{2}{3})$

Step 6.4.2.1.4.2

Cancel the common factor.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{\frac{2 (9 π - 4)}{2 \cdot 2}}{3} + \frac{2}{3})$

Step 6.4.2.1.4.3

Rewrite the expression.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{\frac{9 π - 4}{2}}{3} + \frac{2}{3})$

Step 6.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4}{2} \cdot \frac{1}{3} + \frac{2}{3})$

Step 6.4.2.1.6

Multiply $\frac{9 π - 4}{2} \cdot \frac{1}{3}$ .

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Step 6.4.2.1.6.1

Multiply $\frac{9 π - 4}{2}$ by $\frac{1}{3}$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4}{2 \cdot 3} + \frac{2}{3})$

Step 6.4.2.1.6.2

Multiply $2$ by $3$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4}{6} + \frac{2}{3})$

Step 6.4.2.2

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

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4}{6} + \frac{2}{3} \cdot \frac{2}{2})$

Step 6.4.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.3.1

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

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4}{6} + \frac{2 \cdot 2}{3 \cdot 2})$

Step 6.4.2.3.2

Multiply $3$ by $2$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4}{6} + \frac{2 \cdot 2}{6})$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4 + 2 \cdot 2}{6})$

Step 6.4.2.5

Simplify the numerator.

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Step 6.4.2.5.1

Multiply $2$ by $2$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π - 4 + 4}{6})$

Step 6.4.2.5.2

Add $- 4$ and $4$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π + 0}{6})$

Step 6.4.2.5.3

Add $9 π$ and $0$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{9 π}{6})$

Step 6.4.2.6

Cancel the common factor of $9$ and $6$ .

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Step 6.4.2.6.1

Factor $3$ out of $9 π$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 (3 π)}{6})$

Step 6.4.2.6.2

Cancel the common factors.

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Step 6.4.2.6.2.1

Factor $3$ out of $6$ .

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 (3 π)}{3 (2)})$

Step 6.4.2.6.2.2

Cancel the common factor.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 (3 π)}{3 \cdot 2})$

Step 6.4.2.6.2.3

Rewrite the expression.

$f (\frac{27 π}{4} - 3) = 8 \sin (\frac{3 π}{2})$

Step 6.4.2.7

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{27 π}{4} - 3) = 8 (- \sin (\frac{π}{2}))$

Step 6.4.2.8

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

$f (\frac{27 π}{4} - 3) = 8 (- 1 \cdot 1)$

Step 6.4.2.9

Multiply $8 (- 1 \cdot 1)$ .

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Step 6.4.2.9.1

Multiply $- 1$ by $1$ .

$f (\frac{27 π}{4} - 3) = 8 \cdot - 1$

Step 6.4.2.9.2

Multiply $8$ by $- 1$ .

$f (\frac{27 π}{4} - 3) = - 8$

Step 6.4.2.10

The final answer is $- 8$ .

$- 8$

Step 6.5

Find the point at $x = 9 π - 3$ .

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Step 6.5.1

Replace the variable $x$ with $9 π - 3$ in the expression.

$f (9 π - 3) = 8 \sin (\frac{2 (9 π - 3)}{9} + \frac{2}{3})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify terms.

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Step 6.5.2.1.1

Cancel the common factor of $9 π - 3$ and $9$ .

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Step 6.5.2.1.1.1

Factor $3$ out of $2 (9 π - 3)$ .

$f (9 π - 3) = 8 \sin (\frac{3 (2 (3 π - 1))}{9} + \frac{2}{3})$

Step 6.5.2.1.1.2

Cancel the common factors.

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Step 6.5.2.1.1.2.1

Factor $3$ out of $9$ .

$f (9 π - 3) = 8 \sin (\frac{3 (2 (3 π - 1))}{3 (3)} + \frac{2}{3})$

Step 6.5.2.1.1.2.2

Cancel the common factor.

$f (9 π - 3) = 8 \sin (\frac{3 (2 (3 π - 1))}{3 \cdot 3} + \frac{2}{3})$

Step 6.5.2.1.1.2.3

Rewrite the expression.

$f (9 π - 3) = 8 \sin (\frac{2 (3 π - 1)}{3} + \frac{2}{3})$

Step 6.5.2.1.2

Combine the numerators over the common denominator.

$f (9 π - 3) = 8 \sin (\frac{2 (3 π - 1) + 2}{3})$

Step 6.5.2.2

Simplify each term.

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Step 6.5.2.2.1

Apply the distributive property.

$f (9 π - 3) = 8 \sin (\frac{2 (3 π) + 2 \cdot - 1 + 2}{3})$

Step 6.5.2.2.2

Multiply $3$ by $2$ .

$f (9 π - 3) = 8 \sin (\frac{6 π + 2 \cdot - 1 + 2}{3})$

Step 6.5.2.2.3

Multiply $2$ by $- 1$ .

$f (9 π - 3) = 8 \sin (\frac{6 π - 2 + 2}{3})$

Step 6.5.2.3

Reduce the expression by cancelling the common factors.

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Step 6.5.2.3.1

Add $- 2$ and $2$ .

$f (9 π - 3) = 8 \sin (\frac{6 π + 0}{3})$

Step 6.5.2.3.2

Add $6 π$ and $0$ .

$f (9 π - 3) = 8 \sin (\frac{6 π}{3})$

Step 6.5.2.3.3

Cancel the common factor of $6$ and $3$ .

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Step 6.5.2.3.3.1

Factor $3$ out of $6 π$ .

$f (9 π - 3) = 8 \sin (\frac{3 (2 π)}{3})$

Step 6.5.2.3.3.2

Cancel the common factors.

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Step 6.5.2.3.3.2.1

Factor $3$ out of $3$ .

$f (9 π - 3) = 8 \sin (\frac{3 (2 π)}{3 (1)})$

Step 6.5.2.3.3.2.2

Cancel the common factor.

$f (9 π - 3) = 8 \sin (\frac{3 (2 π)}{3 \cdot 1})$

Step 6.5.2.3.3.2.3

Rewrite the expression.

$f (9 π - 3) = 8 \sin (\frac{2 π}{1})$

Step 6.5.2.3.3.2.4

Divide $2 π$ by $1$ .

$f (9 π - 3) = 8 \sin (2 π)$

Step 6.5.2.4

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (9 π - 3) = 8 \sin (0)$

Step 6.5.2.5

The exact value of $\sin (0)$ is $0$ .

$f (9 π - 3) = 8 \cdot 0$

Step 6.5.2.6

Multiply $8$ by $0$ .

$f (9 π - 3) = 0$

Step 6.5.2.7

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 3 & 0 \\ \frac{9 π}{4} - 3 & 8 \\ \frac{9 π}{2} - 3 & 0 \\ \frac{27 π}{4} - 3 & - 8 \\ 9 π - 3 & 0 \end{array}$

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: $8$

Period: $9 π$

Phase Shift: $- 3$ ( $3$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 3 & 0 \\ \frac{9 π}{4} - 3 & 8 \\ \frac{9 π}{2} - 3 & 0 \\ \frac{27 π}{4} - 3 & - 8 \\ 9 π - 3 & 0 \end{array}$

Step 8