Trigonometry Examples

$y = - 4 \cos (\frac{1}{2} \cdot (x - \frac{π}{3}) + 3)$

Step 1

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

$a = - 4$

$b = \frac{1}{2}$

$c = - (- \frac{π}{6} + 3)$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

Find the period of $- 4 \cos (\frac{x}{2} - \frac{π}{6} + 3)$ .

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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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.5

Multiply $2$ by $2$ .

$4 π$

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: $\frac{- (- \frac{π}{6} + 3)}{\frac{1}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- (- \frac{π}{6} + 3) \cdot 2$

Step 4.4

Apply the distributive property.

Phase Shift: $(\frac{π}{6} - 1 \cdot 3) \cdot 2$

Step 4.5

Multiply $- - \frac{π}{6}$ .

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

Phase Shift: $(1 (\frac{π}{6}) - 1 \cdot 3) \cdot 2$

Step 4.5.2

Multiply $\frac{π}{6}$ by $1$ .

Phase Shift: $(\frac{π}{6} - 1 \cdot 3) \cdot 2$

Step 4.6

Multiply $- 1$ by $3$ .

Phase Shift: $(\frac{π}{6} - 3) \cdot 2$

Step 4.7

Apply the distributive property.

Phase Shift: $\frac{π}{6} \cdot 2 - 3 \cdot 2$

Step 4.8

Cancel the common factor of $2$ .

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Step 4.8.1

Factor $2$ out of $6$ .

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

Step 4.8.2

Cancel the common factor.

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

Step 4.8.3

Rewrite the expression.

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

Step 4.9

Multiply $- 3$ by $2$ .

Phase Shift: $\frac{π}{3} - 6$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $4 π$

Phase Shift: $\frac{π}{3} - 6$ ( $- (\frac{π}{3} - 6)$ 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 $x = \frac{π}{3} - 6$ .

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Step 6.1.1

Replace the variable $x$ with $\frac{π}{3} - 6$ in the expression.

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

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

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.1.2.1.1.2

Combine $- 6$ and $\frac{3}{3}$ .

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

Step 6.1.2.1.1.3

Combine the numerators over the common denominator.

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

Step 6.1.2.1.1.4

Multiply $- 6$ by $3$ .

$f (\frac{π}{3} - 6) = - 4 \cos (\frac{\frac{π - 18}{3}}{2} - \frac{π}{6} + 3)$

Step 6.1.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{π}{3} - 6) = - 4 \cos (\frac{π - 18}{3} \cdot \frac{1}{2} - \frac{π}{6} + 3)$

Step 6.1.2.1.3

Multiply $\frac{π - 18}{3} \cdot \frac{1}{2}$ .

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Step 6.1.2.1.3.1

Multiply $\frac{π - 18}{3}$ by $\frac{1}{2}$ .

$f (\frac{π}{3} - 6) = - 4 \cos (\frac{π - 18}{3 \cdot 2} - \frac{π}{6} + 3)$

Step 6.1.2.1.3.2

Multiply $3$ by $2$ .

$f (\frac{π}{3} - 6) = - 4 \cos (\frac{π - 18}{6} - \frac{π}{6} + 3)$

Step 6.1.2.2

Simplify terms.

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Step 6.1.2.2.1

Combine the numerators over the common denominator.

$f (\frac{π}{3} - 6) = - 4 \cos (3 + \frac{π - 18 - π}{6})$

Step 6.1.2.2.2

Subtract $π$ from $π$ .

$f (\frac{π}{3} - 6) = - 4 \cos (3 + \frac{0 - 18}{6})$

Step 6.1.2.2.3

Simplify the expression.

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Step 6.1.2.2.3.1

Subtract $18$ from $0$ .

$f (\frac{π}{3} - 6) = - 4 \cos (3 + \frac{- 18}{6})$

Step 6.1.2.2.3.2

Divide $- 18$ by $6$ .

$f (\frac{π}{3} - 6) = - 4 \cos (3 - 3)$

Step 6.1.2.2.3.3

Subtract $3$ from $3$ .

$f (\frac{π}{3} - 6) = - 4 \cos (0)$

Step 6.1.2.3

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{3} - 6) = - 4 \cdot 1$

Step 6.1.2.4

Multiply $- 4$ by $1$ .

$f (\frac{π}{3} - 6) = - 4$

Step 6.1.2.5

The final answer is $- 4$ .

$- 4$

Step 6.2

Find the point at $x = \frac{4 π}{3} - 6$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{4 π}{3} - 6$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Cancel the common factor of $\frac{4 π}{3} - 6$ and $2$ .

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Step 6.2.2.1.1

Factor $2$ out of $\frac{4 π}{3}$ .

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

Step 6.2.2.1.2

Factor $2$ out of $- 6$ .

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

Step 6.2.2.1.3

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

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

Step 6.2.2.1.4

Cancel the common factors.

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Step 6.2.2.1.4.1

Factor $2$ out of $2$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{2 π}{3} - 3)}{2 (1)} - \frac{π}{6} + 3)$

Step 6.2.2.1.4.2

Cancel the common factor.

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{2 π}{3} - 3)}{2 \cdot 1} - \frac{π}{6} + 3)$

Step 6.2.2.1.4.3

Rewrite the expression.

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{\frac{2 π}{3} - 3}{1} - \frac{π}{6} + 3)$

Step 6.2.2.1.4.4

Divide $\frac{2 π}{3} - 3$ by $1$ .

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

Step 6.2.2.2

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

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

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 $\frac{2 π}{3}$ by $\frac{2}{2}$ .

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

Step 6.2.2.3.2

Multiply $3$ by $2$ .

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

Step 6.2.2.4

Combine the numerators over the common denominator.

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

Step 6.2.2.5

Find the common denominator.

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Step 6.2.2.5.1

Write $- 3$ as a fraction with denominator $1$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{2 π \cdot 2 - π}{6} + \frac{- 3}{1} + 3)$

Step 6.2.2.5.2

Multiply $\frac{- 3}{1}$ by $\frac{6}{6}$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{2 π \cdot 2 - π}{6} + \frac{- 3}{1} \cdot \frac{6}{6} + 3)$

Step 6.2.2.5.3

Multiply $\frac{- 3}{1}$ by $\frac{6}{6}$ .

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

Step 6.2.2.5.4

Write $3$ as a fraction with denominator $1$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{2 π \cdot 2 - π}{6} + \frac{- 3 \cdot 6}{6} + \frac{3}{1})$

Step 6.2.2.5.5

Multiply $\frac{3}{1}$ by $\frac{6}{6}$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{2 π \cdot 2 - π}{6} + \frac{- 3 \cdot 6}{6} + \frac{3}{1} \cdot \frac{6}{6})$

Step 6.2.2.5.6

Multiply $\frac{3}{1}$ by $\frac{6}{6}$ .

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

Step 6.2.2.6

Combine the numerators over the common denominator.

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

Step 6.2.2.7

Simplify each term.

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Step 6.2.2.7.1

Multiply $2$ by $2$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{4 π - π - 3 \cdot 6 + 3 \cdot 6}{6})$

Step 6.2.2.7.2

Multiply $- 3$ by $6$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{4 π - π - 18 + 3 \cdot 6}{6})$

Step 6.2.2.7.3

Multiply $3$ by $6$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{4 π - π - 18 + 18}{6})$

Step 6.2.2.8

Simplify terms.

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Step 6.2.2.8.1

Subtract $π$ from $4 π$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{3 π - 18 + 18}{6})$

Step 6.2.2.8.2

Simplify by adding numbers.

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Step 6.2.2.8.2.1

Add $- 18$ and $18$ .

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

Step 6.2.2.8.2.2

Add $3 π$ and $0$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{3 π}{6})$

Step 6.2.2.8.3

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

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Step 6.2.2.8.3.1

Factor $3$ out of $3 π$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{3 (π)}{6})$

Step 6.2.2.8.3.2

Cancel the common factors.

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Step 6.2.2.8.3.2.1

Factor $3$ out of $6$ .

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.8.3.2.2

Cancel the common factor.

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.8.3.2.3

Rewrite the expression.

$f (\frac{4 π}{3} - 6) = - 4 \cos (\frac{π}{2})$

Step 6.2.2.9

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{4 π}{3} - 6) = - 4 \cdot 0$

Step 6.2.2.10

Multiply $- 4$ by $0$ .

$f (\frac{4 π}{3} - 6) = 0$

Step 6.2.2.11

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{7 π}{3} - 6$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{7 π}{3} - 6$ in the expression.

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{\frac{7 π}{3} - 6}{2} - \frac{π}{6} + 3)$

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

Simplify the numerator.

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Step 6.3.2.1.1.1

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{\frac{7 π}{3} - 6 \cdot \frac{3}{3}}{2} - \frac{π}{6} + 3)$

Step 6.3.2.1.1.2

Combine $- 6$ and $\frac{3}{3}$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{\frac{7 π}{3} + \frac{- 6 \cdot 3}{3}}{2} - \frac{π}{6} + 3)$

Step 6.3.2.1.1.3

Combine the numerators over the common denominator.

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{\frac{7 π - 6 \cdot 3}{3}}{2} - \frac{π}{6} + 3)$

Step 6.3.2.1.1.4

Multiply $- 6$ by $3$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{\frac{7 π - 18}{3}}{2} - \frac{π}{6} + 3)$

Step 6.3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{7 π - 18}{3} \cdot \frac{1}{2} - \frac{π}{6} + 3)$

Step 6.3.2.1.3

Multiply $\frac{7 π - 18}{3} \cdot \frac{1}{2}$ .

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Step 6.3.2.1.3.1

Multiply $\frac{7 π - 18}{3}$ by $\frac{1}{2}$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{7 π - 18}{3 \cdot 2} - \frac{π}{6} + 3)$

Step 6.3.2.1.3.2

Multiply $3$ by $2$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (\frac{7 π - 18}{6} - \frac{π}{6} + 3)$

Step 6.3.2.2

Simplify terms.

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Step 6.3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{7 π - 18 - π}{6})$

Step 6.3.2.2.2

Subtract $π$ from $7 π$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{6 π - 18}{6})$

Step 6.3.2.2.3

Cancel the common factor of $6 π - 18$ and $6$ .

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Step 6.3.2.2.3.1

Factor $6$ out of $6 π$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{6 (π) - 18}{6})$

Step 6.3.2.2.3.2

Factor $6$ out of $- 18$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{6 (π) + 6 \cdot - 3}{6})$

Step 6.3.2.2.3.3

Factor $6$ out of $6 (π) + 6 (- 3)$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{6 (π - 3)}{6})$

Step 6.3.2.2.3.4

Cancel the common factors.

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Step 6.3.2.2.3.4.1

Factor $6$ out of $6$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{6 (π - 3)}{6 (1)})$

Step 6.3.2.2.3.4.2

Cancel the common factor.

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{6 (π - 3)}{6 \cdot 1})$

Step 6.3.2.2.3.4.3

Rewrite the expression.

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + \frac{π - 3}{1})$

Step 6.3.2.2.3.4.4

Divide $π - 3$ by $1$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (3 + π - 3)$

Step 6.3.2.2.4

Simplify by subtracting numbers.

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Step 6.3.2.2.4.1

Subtract $3$ from $3$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (0 + π)$

Step 6.3.2.2.4.2

Add $0$ and $π$ .

$f (\frac{7 π}{3} - 6) = - 4 \cos (π)$

Step 6.3.2.3

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

$f (\frac{7 π}{3} - 6) = - 4 (- \cos (0))$

Step 6.3.2.4

The exact value of $\cos (0)$ is $1$ .

$f (\frac{7 π}{3} - 6) = - 4 (- 1 \cdot 1)$

Step 6.3.2.5

Multiply $- 4 (- 1 \cdot 1)$ .

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Step 6.3.2.5.1

Multiply $- 1$ by $1$ .

$f (\frac{7 π}{3} - 6) = - 4 \cdot - 1$

Step 6.3.2.5.2

Multiply $- 4$ by $- 1$ .

$f (\frac{7 π}{3} - 6) = 4$

Step 6.3.2.6

The final answer is $4$ .

$4$

Step 6.4

Find the point at $x = \frac{10 π}{3} - 6$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{10 π}{3} - 6$ in the expression.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{\frac{10 π}{3} - 6}{2} - \frac{π}{6} + 3)$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $\frac{10 π}{3} - 6$ and $2$ .

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Step 6.4.2.1.1

Factor $2$ out of $\frac{10 π}{3}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{5 π}{3}) - 6}{2} - \frac{π}{6} + 3)$

Step 6.4.2.1.2

Factor $2$ out of $- 6$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{5 π}{3}) + 2 \cdot - 3}{2} - \frac{π}{6} + 3)$

Step 6.4.2.1.3

Factor $2$ out of $2 \frac{5 π}{3} + 2 (- 3)$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{5 π}{3} - 3)}{2} - \frac{π}{6} + 3)$

Step 6.4.2.1.4

Cancel the common factors.

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Step 6.4.2.1.4.1

Factor $2$ out of $2$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{5 π}{3} - 3)}{2 (1)} - \frac{π}{6} + 3)$

Step 6.4.2.1.4.2

Cancel the common factor.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{2 (\frac{5 π}{3} - 3)}{2 \cdot 1} - \frac{π}{6} + 3)$

Step 6.4.2.1.4.3

Rewrite the expression.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{\frac{5 π}{3} - 3}{1} - \frac{π}{6} + 3)$

Step 6.4.2.1.4.4

Divide $\frac{5 π}{3} - 3$ by $1$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π}{3} - 3 - \frac{π}{6} + 3)$

Step 6.4.2.2

To write $\frac{5 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π}{3} \cdot \frac{2}{2} - \frac{π}{6} - 3 + 3)$

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 $\frac{5 π}{3}$ by $\frac{2}{2}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2}{3 \cdot 2} - \frac{π}{6} - 3 + 3)$

Step 6.4.2.3.2

Multiply $3$ by $2$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2}{6} - \frac{π}{6} - 3 + 3)$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} - 3 + 3)$

Step 6.4.2.5

Find the common denominator.

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Step 6.4.2.5.1

Write $- 3$ as a fraction with denominator $1$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} + \frac{- 3}{1} + 3)$

Step 6.4.2.5.2

Multiply $\frac{- 3}{1}$ by $\frac{6}{6}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} + \frac{- 3}{1} \cdot \frac{6}{6} + 3)$

Step 6.4.2.5.3

Multiply $\frac{- 3}{1}$ by $\frac{6}{6}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} + \frac{- 3 \cdot 6}{6} + 3)$

Step 6.4.2.5.4

Write $3$ as a fraction with denominator $1$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} + \frac{- 3 \cdot 6}{6} + \frac{3}{1})$

Step 6.4.2.5.5

Multiply $\frac{3}{1}$ by $\frac{6}{6}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} + \frac{- 3 \cdot 6}{6} + \frac{3}{1} \cdot \frac{6}{6})$

Step 6.4.2.5.6

Multiply $\frac{3}{1}$ by $\frac{6}{6}$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π}{6} + \frac{- 3 \cdot 6}{6} + \frac{3 \cdot 6}{6})$

Step 6.4.2.6

Combine the numerators over the common denominator.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{5 π \cdot 2 - π - 3 \cdot 6 + 3 \cdot 6}{6})$

Step 6.4.2.7

Simplify each term.

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Step 6.4.2.7.1

Multiply $2$ by $5$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{10 π - π - 3 \cdot 6 + 3 \cdot 6}{6})$

Step 6.4.2.7.2

Multiply $- 3$ by $6$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{10 π - π - 18 + 3 \cdot 6}{6})$

Step 6.4.2.7.3

Multiply $3$ by $6$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{10 π - π - 18 + 18}{6})$

Step 6.4.2.8

Simplify terms.

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Step 6.4.2.8.1

Subtract $π$ from $10 π$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{9 π - 18 + 18}{6})$

Step 6.4.2.8.2

Simplify by adding numbers.

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Step 6.4.2.8.2.1

Add $- 18$ and $18$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{9 π + 0}{6})$

Step 6.4.2.8.2.2

Add $9 π$ and $0$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{9 π}{6})$

Step 6.4.2.8.3

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

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Step 6.4.2.8.3.1

Factor $3$ out of $9 π$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{3 (3 π)}{6})$

Step 6.4.2.8.3.2

Cancel the common factors.

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Step 6.4.2.8.3.2.1

Factor $3$ out of $6$ .

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{3 (3 π)}{3 (2)})$

Step 6.4.2.8.3.2.2

Cancel the common factor.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{3 (3 π)}{3 \cdot 2})$

Step 6.4.2.8.3.2.3

Rewrite the expression.

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{3 π}{2})$

Step 6.4.2.9

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

$f (\frac{10 π}{3} - 6) = - 4 \cos (\frac{π}{2})$

Step 6.4.2.10

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{10 π}{3} - 6) = - 4 \cdot 0$

Step 6.4.2.11

Multiply $- 4$ by $0$ .

$f (\frac{10 π}{3} - 6) = 0$

Step 6.4.2.12

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{13 π}{3} - 6$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{13 π}{3} - 6$ in the expression.

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{\frac{13 π}{3} - 6}{2} - \frac{π}{6} + 3)$

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

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{\frac{13 π}{3} - 6 \cdot \frac{3}{3}}{2} - \frac{π}{6} + 3)$

Step 6.5.2.1.1.2

Combine $- 6$ and $\frac{3}{3}$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{\frac{13 π}{3} + \frac{- 6 \cdot 3}{3}}{2} - \frac{π}{6} + 3)$

Step 6.5.2.1.1.3

Combine the numerators over the common denominator.

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{\frac{13 π - 6 \cdot 3}{3}}{2} - \frac{π}{6} + 3)$

Step 6.5.2.1.1.4

Multiply $- 6$ by $3$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{\frac{13 π - 18}{3}}{2} - \frac{π}{6} + 3)$

Step 6.5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{13 π - 18}{3} \cdot \frac{1}{2} - \frac{π}{6} + 3)$

Step 6.5.2.1.3

Multiply $\frac{13 π - 18}{3} \cdot \frac{1}{2}$ .

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Step 6.5.2.1.3.1

Multiply $\frac{13 π - 18}{3}$ by $\frac{1}{2}$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{13 π - 18}{3 \cdot 2} - \frac{π}{6} + 3)$

Step 6.5.2.1.3.2

Multiply $3$ by $2$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (\frac{13 π - 18}{6} - \frac{π}{6} + 3)$

Step 6.5.2.2

Simplify terms.

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Step 6.5.2.2.1

Combine the numerators over the common denominator.

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{13 π - 18 - π}{6})$

Step 6.5.2.2.2

Subtract $π$ from $13 π$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{12 π - 18}{6})$

Step 6.5.2.2.3

Cancel the common factor of $12 π - 18$ and $6$ .

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Step 6.5.2.2.3.1

Factor $6$ out of $12 π$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{6 (2 π) - 18}{6})$

Step 6.5.2.2.3.2

Factor $6$ out of $- 18$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{6 (2 π) + 6 \cdot - 3}{6})$

Step 6.5.2.2.3.3

Factor $6$ out of $6 (2 π) + 6 (- 3)$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{6 (2 π - 3)}{6})$

Step 6.5.2.2.3.4

Cancel the common factors.

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Step 6.5.2.2.3.4.1

Factor $6$ out of $6$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{6 (2 π - 3)}{6 (1)})$

Step 6.5.2.2.3.4.2

Cancel the common factor.

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{6 (2 π - 3)}{6 \cdot 1})$

Step 6.5.2.2.3.4.3

Rewrite the expression.

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + \frac{2 π - 3}{1})$

Step 6.5.2.2.3.4.4

Divide $2 π - 3$ by $1$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (3 + 2 π - 3)$

Step 6.5.2.2.4

Simplify by subtracting numbers.

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Step 6.5.2.2.4.1

Subtract $3$ from $3$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (2 π + 0)$

Step 6.5.2.2.4.2

Add $2 π$ and $0$ .

$f (\frac{13 π}{3} - 6) = - 4 \cos (2 π)$

Step 6.5.2.3

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

$f (\frac{13 π}{3} - 6) = - 4 \cos (0)$

Step 6.5.2.4

The exact value of $\cos (0)$ is $1$ .

$f (\frac{13 π}{3} - 6) = - 4 \cdot 1$

Step 6.5.2.5

Multiply $- 4$ by $1$ .

$f (\frac{13 π}{3} - 6) = - 4$

Step 6.5.2.6

The final answer is $- 4$ .

$- 4$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{3} - 6 & - 4 \\ \frac{4 π}{3} - 6 & 0 \\ \frac{7 π}{3} - 6 & 4 \\ \frac{10 π}{3} - 6 & 0 \\ \frac{13 π}{3} - 6 & - 4 \end{array}$

Step 7

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

Amplitude: $4$

Period: $4 π$

Phase Shift: $\frac{π}{3} - 6$ ( $- (\frac{π}{3} - 6)$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{π}{3} - 6 & - 4 \\ \frac{4 π}{3} - 6 & 0 \\ \frac{7 π}{3} - 6 & 4 \\ \frac{10 π}{3} - 6 & 0 \\ \frac{13 π}{3} - 6 & - 4 \end{array}$

Step 8