Trigonometry Examples

$y = 3 \cos (2 (x + \frac{π}{3}))$

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.

$a = 3$

$b = 2$

$c = - \frac{2 π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.4.2

Divide $π$ by $1$ .

$π$

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: $\frac{- \frac{2 π}{3}}{2}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

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

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

Step 4.4.2

Factor $2$ out of $- 2 π$ .

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

Step 4.4.3

Cancel the common factor.

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

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- π}{3}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

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

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Step 6.1.1

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

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

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

Multiply $2 (- \frac{π}{3})$ .

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Step 6.1.2.1.1.1

Multiply $- 1$ by $2$ .

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

Step 6.1.2.1.1.2

Combine $- 2$ and $\frac{π}{3}$ .

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

Step 6.1.2.1.2

Move the negative in front of the fraction.

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

Step 6.1.2.2

Simplify terms.

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Step 6.1.2.2.1

Combine the numerators over the common denominator.

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

Step 6.1.2.2.2

Add $- 2 π$ and $2 π$ .

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

Step 6.1.2.2.3

Divide $0$ by $3$ .

$f (- \frac{π}{3}) = 3 \cos (0)$

Step 6.1.2.3

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

$f (- \frac{π}{3}) = 3 \cdot 1$

Step 6.1.2.4

Multiply $3$ by $1$ .

$f (- \frac{π}{3}) = 3$

Step 6.1.2.5

The final answer is $3$ .

$3$

Step 6.2

Find the point at $x = - \frac{π}{12}$ .

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Step 6.2.1

Replace the variable $x$ with $- \frac{π}{12}$ in the expression.

$f (- \frac{π}{12}) = 3 \cos (2 (- \frac{π}{12}) + \frac{2 π}{3})$

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 $2$ .

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Step 6.2.2.1.1.1

Move the leading negative in $- \frac{π}{12}$ into the numerator.

$f (- \frac{π}{12}) = 3 \cos (2 (\frac{- π}{12}) + \frac{2 π}{3})$

Step 6.2.2.1.1.2

Factor $2$ out of $12$ .

$f (- \frac{π}{12}) = 3 \cos (2 (\frac{- π}{2 (6)}) + \frac{2 π}{3})$

Step 6.2.2.1.1.3

Cancel the common factor.

$f (- \frac{π}{12}) = 3 \cos (2 (\frac{- π}{2 \cdot 6}) + \frac{2 π}{3})$

Step 6.2.2.1.1.4

Rewrite the expression.

$f (- \frac{π}{12}) = 3 \cos (\frac{- π}{6} + \frac{2 π}{3})$

Step 6.2.2.1.2

Move the negative in front of the fraction.

$f (- \frac{π}{12}) = 3 \cos (- \frac{π}{6} + \frac{2 π}{3})$

Step 6.2.2.2

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

$f (- \frac{π}{12}) = 3 \cos (- \frac{π}{6} + \frac{2 π}{3} \cdot \frac{2}{2})$

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 $\frac{2 π}{3}$ by $\frac{2}{2}$ .

$f (- \frac{π}{12}) = 3 \cos (- \frac{π}{6} + \frac{2 π \cdot 2}{3 \cdot 2})$

Step 6.2.2.3.2

Multiply $3$ by $2$ .

$f (- \frac{π}{12}) = 3 \cos (- \frac{π}{6} + \frac{2 π \cdot 2}{6})$

Step 6.2.2.4

Combine the numerators over the common denominator.

$f (- \frac{π}{12}) = 3 \cos (\frac{- π + 2 π \cdot 2}{6})$

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Multiply $2$ by $2$ .

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

Step 6.2.2.5.2

Add $- π$ and $4 π$ .

$f (- \frac{π}{12}) = 3 \cos (\frac{3 π}{6})$

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 $3 π$ .

$f (- \frac{π}{12}) = 3 \cos (\frac{3 (π)}{6})$

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 $6$ .

$f (- \frac{π}{12}) = 3 \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.6.2.2

Cancel the common factor.

$f (- \frac{π}{12}) = 3 \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.6.2.3

Rewrite the expression.

$f (- \frac{π}{12}) = 3 \cos (\frac{π}{2})$

Step 6.2.2.7

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

$f (- \frac{π}{12}) = 3 \cdot 0$

Step 6.2.2.8

Multiply $3$ by $0$ .

$f (- \frac{π}{12}) = 0$

Step 6.2.2.9

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{π}{6}$ .

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Step 6.3.1

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

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Factor $2$ out of $6$ .

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

Step 6.3.2.1.2

Cancel the common factor.

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

Step 6.3.2.1.3

Rewrite the expression.

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

Step 6.3.2.2

Combine the numerators over the common denominator.

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

Step 6.3.2.3

Add $π$ and $2 π$ .

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

Step 6.3.2.4

Cancel the common factor of $3$ .

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Step 6.3.2.4.1

Cancel the common factor.

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

Step 6.3.2.4.2

Divide $π$ by $1$ .

$f (\frac{π}{6}) = 3 \cos (π)$

Step 6.3.2.5

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.

$f (\frac{π}{6}) = 3 (- \cos (0))$

Step 6.3.2.6

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

$f (\frac{π}{6}) = 3 (- 1 \cdot 1)$

Step 6.3.2.7

Multiply $3 (- 1 \cdot 1)$ .

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Step 6.3.2.7.1

Multiply $- 1$ by $1$ .

$f (\frac{π}{6}) = 3 \cdot - 1$

Step 6.3.2.7.2

Multiply $3$ by $- 1$ .

$f (\frac{π}{6}) = - 3$

Step 6.3.2.8

The final answer is $- 3$ .

$- 3$

Step 6.4

Find the point at $x = \frac{5 π}{12}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{5 π}{12}$ in the expression.

$f (\frac{5 π}{12}) = 3 \cos (2 (\frac{5 π}{12}) + \frac{2 π}{3})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $2$ .

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Step 6.4.2.1.1

Factor $2$ out of $12$ .

$f (\frac{5 π}{12}) = 3 \cos (2 (\frac{5 π}{2 (6)}) + \frac{2 π}{3})$

Step 6.4.2.1.2

Cancel the common factor.

$f (\frac{5 π}{12}) = 3 \cos (2 (\frac{5 π}{2 \cdot 6}) + \frac{2 π}{3})$

Step 6.4.2.1.3

Rewrite the expression.

$f (\frac{5 π}{12}) = 3 \cos (\frac{5 π}{6} + \frac{2 π}{3})$

Step 6.4.2.2

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

$f (\frac{5 π}{12}) = 3 \cos (\frac{5 π}{6} + \frac{2 π}{3} \cdot \frac{2}{2})$

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 $\frac{2 π}{3}$ by $\frac{2}{2}$ .

$f (\frac{5 π}{12}) = 3 \cos (\frac{5 π}{6} + \frac{2 π \cdot 2}{3 \cdot 2})$

Step 6.4.2.3.2

Multiply $3$ by $2$ .

$f (\frac{5 π}{12}) = 3 \cos (\frac{5 π}{6} + \frac{2 π \cdot 2}{6})$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (\frac{5 π}{12}) = 3 \cos (\frac{5 π + 2 π \cdot 2}{6})$

Step 6.4.2.5

Simplify the numerator.

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Step 6.4.2.5.1

Multiply $2$ by $2$ .

$f (\frac{5 π}{12}) = 3 \cos (\frac{5 π + 4 π}{6})$

Step 6.4.2.5.2

Add $5 π$ and $4 π$ .

$f (\frac{5 π}{12}) = 3 \cos (\frac{9 π}{6})$

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 $9 π$ .

$f (\frac{5 π}{12}) = 3 \cos (\frac{3 (3 π)}{6})$

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 $6$ .

$f (\frac{5 π}{12}) = 3 \cos (\frac{3 (3 π)}{3 (2)})$

Step 6.4.2.6.2.2

Cancel the common factor.

$f (\frac{5 π}{12}) = 3 \cos (\frac{3 (3 π)}{3 \cdot 2})$

Step 6.4.2.6.2.3

Rewrite the expression.

$f (\frac{5 π}{12}) = 3 \cos (\frac{3 π}{2})$

Step 6.4.2.7

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

$f (\frac{5 π}{12}) = 3 \cos (\frac{π}{2})$

Step 6.4.2.8

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

$f (\frac{5 π}{12}) = 3 \cdot 0$

Step 6.4.2.9

Multiply $3$ by $0$ .

$f (\frac{5 π}{12}) = 0$

Step 6.4.2.10

The final answer is $0$ .

$0$

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Multiply $2 (\frac{2 π}{3})$ .

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Step 6.5.2.1.1

Combine $2$ and $\frac{2 π}{3}$ .

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

Step 6.5.2.1.2

Multiply $2$ by $2$ .

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

Step 6.5.2.2

Combine the numerators over the common denominator.

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

Step 6.5.2.3

Add $4 π$ and $2 π$ .

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

Step 6.5.2.4

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

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Step 6.5.2.4.1

Factor $3$ out of $6 π$ .

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

Step 6.5.2.4.2

Cancel the common factors.

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Step 6.5.2.4.2.1

Factor $3$ out of $3$ .

$f (\frac{2 π}{3}) = 3 \cos (\frac{3 (2 π)}{3 (1)})$

Step 6.5.2.4.2.2

Cancel the common factor.

$f (\frac{2 π}{3}) = 3 \cos (\frac{3 (2 π)}{3 \cdot 1})$

Step 6.5.2.4.2.3

Rewrite the expression.

$f (\frac{2 π}{3}) = 3 \cos (\frac{2 π}{1})$

Step 6.5.2.4.2.4

Divide $2 π$ by $1$ .

$f (\frac{2 π}{3}) = 3 \cos (2 π)$

Step 6.5.2.5

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

$f (\frac{2 π}{3}) = 3 \cos (0)$

Step 6.5.2.6

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

$f (\frac{2 π}{3}) = 3 \cdot 1$

Step 6.5.2.7

Multiply $3$ by $1$ .

$f (\frac{2 π}{3}) = 3$

Step 6.5.2.8

The final answer is $3$ .

$3$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{3} & 3 \\ - \frac{π}{12} & 0 \\ \frac{π}{6} & - 3 \\ \frac{5 π}{12} & 0 \\ \frac{2 π}{3} & 3 \end{array}$

Step 7

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

Amplitude: $3$

Period: $π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{π}{3} & 3 \\ - \frac{π}{12} & 0 \\ \frac{π}{6} & - 3 \\ \frac{5 π}{12} & 0 \\ \frac{2 π}{3} & 3 \end{array}$

Step 8