Trigonometry Examples

$y = - \frac{1}{2} \cdot \sin (3 x - \frac{π}{3}) - 1$

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.

$a = - \frac{1}{2}$

$b = 3$

$c = \frac{π}{3}$

$d = - 1$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

Step 3

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of $- \frac{\sin (3 x - \frac{π}{3})}{2}$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 3.2

Find the period of $- 1$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{2 π}{3}$

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: $\frac{\frac{π}{3}}{3}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{3} \cdot \frac{1}{3}$

Step 4.4

Multiply $\frac{π}{3} \cdot \frac{1}{3}$ .

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Step 4.4.1

Multiply $\frac{π}{3}$ by $\frac{1}{3}$ .

Phase Shift: $\frac{π}{3 \cdot 3}$

Step 4.4.2

Multiply $3$ by $3$ .

Phase Shift: $\frac{π}{9}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{9}$ ( $\frac{π}{9}$ to the right)

Vertical Shift: $- 1$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = \frac{π}{9}$ .

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Simplify the numerator.

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Step 6.1.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1.1.1

Factor $3$ out of $9$ .

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

Step 6.1.2.1.1.1.2

Cancel the common factor.

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

Step 6.1.2.1.1.1.3

Rewrite the expression.

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

Step 6.1.2.1.1.2

Combine the numerators over the common denominator.

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

Step 6.1.2.1.1.3

Subtract $π$ from $π$ .

$f (\frac{π}{9}) = - \frac{\sin (\frac{0}{3})}{2} - 1$

Step 6.1.2.1.1.4

Divide $0$ by $3$ .

$f (\frac{π}{9}) = - \frac{\sin (0)}{2} - 1$

Step 6.1.2.1.1.5

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

$f (\frac{π}{9}) = - \frac{0}{2} - 1$

Step 6.1.2.1.2

Divide $0$ by $2$ .

$f (\frac{π}{9}) = - 0 - 1$

Step 6.1.2.1.3

Multiply $- 1$ by $0$ .

$f (\frac{π}{9}) = 0 - 1$

Step 6.1.2.2

Subtract $1$ from $0$ .

$f (\frac{π}{9}) = - 1$

Step 6.1.2.3

The final answer is $- 1$ .

$- 1$

Step 6.2

Find the point at $x = \frac{5 π}{18}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{5 π}{18}$ in the expression.

$f (\frac{5 π}{18}) = - \frac{\sin (3 (\frac{5 π}{18}) - \frac{π}{3})}{2} - 1$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify the numerator.

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Step 6.2.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.1

Factor $3$ out of $18$ .

$f (\frac{5 π}{18}) = - \frac{\sin (3 (\frac{5 π}{3 (6)}) - \frac{π}{3})}{2} - 1$

Step 6.2.2.1.1.2

Cancel the common factor.

$f (\frac{5 π}{18}) = - \frac{\sin (3 (\frac{5 π}{3 \cdot 6}) - \frac{π}{3})}{2} - 1$

Step 6.2.2.1.1.3

Rewrite the expression.

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{5 π}{6} - \frac{π}{3})}{2} - 1$

Step 6.2.2.1.2

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

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{5 π}{6} - \frac{π}{3} \cdot \frac{2}{2})}{2} - 1$

Step 6.2.2.1.3

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

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Step 6.2.2.1.3.1

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

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{5 π}{6} - \frac{π \cdot 2}{3 \cdot 2})}{2} - 1$

Step 6.2.2.1.3.2

Multiply $3$ by $2$ .

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{5 π}{6} - \frac{π \cdot 2}{6})}{2} - 1$

Step 6.2.2.1.4

Combine the numerators over the common denominator.

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{5 π - π \cdot 2}{6})}{2} - 1$

Step 6.2.2.1.5

Simplify the numerator.

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Step 6.2.2.1.5.1

Multiply $2$ by $- 1$ .

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{5 π - 2 π}{6})}{2} - 1$

Step 6.2.2.1.5.2

Subtract $2 π$ from $5 π$ .

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{3 π}{6})}{2} - 1$

Step 6.2.2.1.6

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

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Step 6.2.2.1.6.1

Factor $3$ out of $3 π$ .

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{3 (π)}{6})}{2} - 1$

Step 6.2.2.1.6.2

Cancel the common factors.

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Step 6.2.2.1.6.2.1

Factor $3$ out of $6$ .

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{3 π}{3 \cdot 2})}{2} - 1$

Step 6.2.2.1.6.2.2

Cancel the common factor.

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{3 π}{3 \cdot 2})}{2} - 1$

Step 6.2.2.1.6.2.3

Rewrite the expression.

$f (\frac{5 π}{18}) = - \frac{\sin (\frac{π}{2})}{2} - 1$

Step 6.2.2.1.7

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

$f (\frac{5 π}{18}) = - \frac{1}{2} - 1$

Step 6.2.2.2

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

$f (\frac{5 π}{18}) = - \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 6.2.2.3

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

$f (\frac{5 π}{18}) = - \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 6.2.2.4

Combine the numerators over the common denominator.

$f (\frac{5 π}{18}) = \frac{- 1 - 1 \cdot 2}{2}$

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Multiply $- 1$ by $2$ .

$f (\frac{5 π}{18}) = \frac{- 1 - 2}{2}$

Step 6.2.2.5.2

Subtract $2$ from $- 1$ .

$f (\frac{5 π}{18}) = \frac{- 3}{2}$

Step 6.2.2.6

Move the negative in front of the fraction.

$f (\frac{5 π}{18}) = - \frac{3}{2}$

Step 6.2.2.7

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

Step 6.3

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

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Step 6.3.1

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

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

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

Simplify the numerator.

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Step 6.3.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.1.1

Factor $3$ out of $9$ .

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

Step 6.3.2.1.1.1.2

Cancel the common factor.

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

Step 6.3.2.1.1.1.3

Rewrite the expression.

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

Step 6.3.2.1.1.2

Combine the numerators over the common denominator.

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

Step 6.3.2.1.1.3

Subtract $π$ from $4 π$ .

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

Step 6.3.2.1.1.4

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.4.1

Cancel the common factor.

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

Step 6.3.2.1.1.4.2

Divide $π$ by $1$ .

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

Step 6.3.2.1.1.5

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

$f (\frac{4 π}{9}) = - \frac{\sin (0)}{2} - 1$

Step 6.3.2.1.1.6

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

$f (\frac{4 π}{9}) = - \frac{0}{2} - 1$

Step 6.3.2.1.2

Divide $0$ by $2$ .

$f (\frac{4 π}{9}) = - 0 - 1$

Step 6.3.2.1.3

Multiply $- 1$ by $0$ .

$f (\frac{4 π}{9}) = 0 - 1$

Step 6.3.2.2

Subtract $1$ from $0$ .

$f (\frac{4 π}{9}) = - 1$

Step 6.3.2.3

The final answer is $- 1$ .

$- 1$

Step 6.4

Find the point at $x = \frac{11 π}{18}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{11 π}{18}$ in the expression.

$f (\frac{11 π}{18}) = - \frac{\sin (3 (\frac{11 π}{18}) - \frac{π}{3})}{2} - 1$

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

Simplify the numerator.

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Step 6.4.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.4.2.1.1.1.1

Factor $3$ out of $18$ .

$f (\frac{11 π}{18}) = - \frac{\sin (3 (\frac{11 π}{3 (6)}) - \frac{π}{3})}{2} - 1$

Step 6.4.2.1.1.1.2

Cancel the common factor.

$f (\frac{11 π}{18}) = - \frac{\sin (3 (\frac{11 π}{3 \cdot 6}) - \frac{π}{3})}{2} - 1$

Step 6.4.2.1.1.1.3

Rewrite the expression.

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{11 π}{6} - \frac{π}{3})}{2} - 1$

Step 6.4.2.1.1.2

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

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{11 π}{6} - \frac{π}{3} \cdot \frac{2}{2})}{2} - 1$

Step 6.4.2.1.1.3

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

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Step 6.4.2.1.1.3.1

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

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{11 π}{6} - \frac{π \cdot 2}{3 \cdot 2})}{2} - 1$

Step 6.4.2.1.1.3.2

Multiply $3$ by $2$ .

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{11 π}{6} - \frac{π \cdot 2}{6})}{2} - 1$

Step 6.4.2.1.1.4

Combine the numerators over the common denominator.

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{11 π - π \cdot 2}{6})}{2} - 1$

Step 6.4.2.1.1.5

Simplify the numerator.

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Step 6.4.2.1.1.5.1

Multiply $2$ by $- 1$ .

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{11 π - 2 π}{6})}{2} - 1$

Step 6.4.2.1.1.5.2

Subtract $2 π$ from $11 π$ .

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{9 π}{6})}{2} - 1$

Step 6.4.2.1.1.6

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

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Step 6.4.2.1.1.6.1

Factor $3$ out of $9 π$ .

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{3 (3 π)}{6})}{2} - 1$

Step 6.4.2.1.1.6.2

Cancel the common factors.

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Step 6.4.2.1.1.6.2.1

Factor $3$ out of $6$ .

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{3 (3 π)}{3 (2)})}{2} - 1$

Step 6.4.2.1.1.6.2.2

Cancel the common factor.

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{3 (3 π)}{3 \cdot 2})}{2} - 1$

Step 6.4.2.1.1.6.2.3

Rewrite the expression.

$f (\frac{11 π}{18}) = - \frac{\sin (\frac{3 π}{2})}{2} - 1$

Step 6.4.2.1.1.7

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.

$f (\frac{11 π}{18}) = - \frac{- \sin (\frac{π}{2})}{2} - 1$

Step 6.4.2.1.1.8

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

$f (\frac{11 π}{18}) = - \frac{- 1 \cdot 1}{2} - 1$

Step 6.4.2.1.1.9

Multiply $- 1$ by $1$ .

$f (\frac{11 π}{18}) = - \frac{- 1}{2} - 1$

Step 6.4.2.1.2

Move the negative in front of the fraction.

$f (\frac{11 π}{18}) = \frac{1}{2} - 1$

Step 6.4.2.1.3

Multiply $- - \frac{1}{2}$ .

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Step 6.4.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (\frac{11 π}{18}) = 1 (\frac{1}{2}) - 1$

Step 6.4.2.1.3.2

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

$f (\frac{11 π}{18}) = \frac{1}{2} - 1$

Step 6.4.2.2

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

$f (\frac{11 π}{18}) = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 6.4.2.3

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

$f (\frac{11 π}{18}) = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (\frac{11 π}{18}) = \frac{1 - 1 \cdot 2}{2}$

Step 6.4.2.5

Simplify the numerator.

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Step 6.4.2.5.1

Multiply $- 1$ by $2$ .

$f (\frac{11 π}{18}) = \frac{1 - 2}{2}$

Step 6.4.2.5.2

Subtract $2$ from $1$ .

$f (\frac{11 π}{18}) = \frac{- 1}{2}$

Step 6.4.2.6

Move the negative in front of the fraction.

$f (\frac{11 π}{18}) = - \frac{1}{2}$

Step 6.4.2.7

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 6.5

Find the point at $x = \frac{7 π}{9}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{7 π}{9}$ in the expression.

$f (\frac{7 π}{9}) = - \frac{\sin (3 (\frac{7 π}{9}) - \frac{π}{3})}{2} - 1$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Simplify the numerator.

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Step 6.5.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1.1.1

Factor $3$ out of $9$ .

$f (\frac{7 π}{9}) = - \frac{\sin (3 (\frac{7 π}{3 (3)}) - \frac{π}{3})}{2} - 1$

Step 6.5.2.1.1.1.2

Cancel the common factor.

$f (\frac{7 π}{9}) = - \frac{\sin (3 (\frac{7 π}{3 \cdot 3}) - \frac{π}{3})}{2} - 1$

Step 6.5.2.1.1.1.3

Rewrite the expression.

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{7 π}{3} - \frac{π}{3})}{2} - 1$

Step 6.5.2.1.1.2

Combine the numerators over the common denominator.

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{7 π - π}{3})}{2} - 1$

Step 6.5.2.1.1.3

Subtract $π$ from $7 π$ .

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{6 π}{3})}{2} - 1$

Step 6.5.2.1.1.4

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

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Step 6.5.2.1.1.4.1

Factor $3$ out of $6 π$ .

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{3 (2 π)}{3})}{2} - 1$

Step 6.5.2.1.1.4.2

Cancel the common factors.

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Step 6.5.2.1.1.4.2.1

Factor $3$ out of $3$ .

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{3 (2 π)}{3 (1)})}{2} - 1$

Step 6.5.2.1.1.4.2.2

Cancel the common factor.

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{3 (2 π)}{3 \cdot 1})}{2} - 1$

Step 6.5.2.1.1.4.2.3

Rewrite the expression.

$f (\frac{7 π}{9}) = - \frac{\sin (\frac{2 π}{1})}{2} - 1$

Step 6.5.2.1.1.4.2.4

Divide $2 π$ by $1$ .

$f (\frac{7 π}{9}) = - \frac{\sin (2 π)}{2} - 1$

Step 6.5.2.1.1.5

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

$f (\frac{7 π}{9}) = - \frac{\sin (0)}{2} - 1$

Step 6.5.2.1.1.6

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

$f (\frac{7 π}{9}) = - \frac{0}{2} - 1$

Step 6.5.2.1.2

Divide $0$ by $2$ .

$f (\frac{7 π}{9}) = - 0 - 1$

Step 6.5.2.1.3

Multiply $- 1$ by $0$ .

$f (\frac{7 π}{9}) = 0 - 1$

Step 6.5.2.2

Subtract $1$ from $0$ .

$f (\frac{7 π}{9}) = - 1$

Step 6.5.2.3

The final answer is $- 1$ .

$- 1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{9} & - 1 \\ \frac{5 π}{18} & - \frac{3}{2} \\ \frac{4 π}{9} & - 1 \\ \frac{11 π}{18} & - \frac{1}{2} \\ \frac{7 π}{9} & - 1 \end{array}$

Step 7

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

Amplitude: $\frac{1}{2}$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{9}$ ( $\frac{π}{9}$ to the right)

Vertical Shift: $- 1$

$\begin{array}{c} x & f (x) \\ \frac{π}{9} & - 1 \\ \frac{5 π}{18} & - \frac{3}{2} \\ \frac{4 π}{9} & - 1 \\ \frac{11 π}{18} & - \frac{1}{2} \\ \frac{7 π}{9} & - 1 \end{array}$

Step 8