Trigonometry Examples

$y = - 3 \cos (6 x + π)$

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.

$a = - 3$

$b = 6$

$c = - π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $- 3 \cos (6 x + π)$ .

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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 $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 3.4

Cancel the common factor of $2$ and $6$ .

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Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{3}$

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: $\frac{- π}{6}$

Step 4.3

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{π}{3}$

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

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Cancel the common factor of $6$ .

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Step 6.1.2.1.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

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

Step 6.1.2.1.2

Cancel the common factor.

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

Step 6.1.2.1.3

Rewrite the expression.

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

Step 6.1.2.2

Add $- π$ and $π$ .

$f (- \frac{π}{6}) = - 3 \cos (0)$

Step 6.1.2.3

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

$f (- \frac{π}{6}) = - 3 \cdot 1$

Step 6.1.2.4

Multiply $- 3$ by $1$ .

$f (- \frac{π}{6}) = - 3$

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.

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

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 $6$ .

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Step 6.2.2.1.1.1

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

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

Step 6.2.2.1.1.2

Factor $6$ out of $12$ .

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

Step 6.2.2.1.1.3

Cancel the common factor.

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

Step 6.2.2.1.1.4

Rewrite the expression.

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

Step 6.2.2.1.2

Move the negative in front of the fraction.

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Combine fractions.

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Step 6.2.2.3.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.2.2.3.2

Combine the numerators over the common denominator.

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

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Move $2$ to the left of $π$ .

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

Step 6.2.2.4.2

Add $- π$ and $2 π$ .

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

Step 6.2.2.5

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

$f (- \frac{π}{12}) = - 3 \cdot 0$

Step 6.2.2.6

Multiply $- 3$ by $0$ .

$f (- \frac{π}{12}) = 0$

Step 6.2.2.7

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = 0$ .

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Step 6.3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 3 \cos (6 (0) + π)$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Multiply $6$ by $0$ .

$f (0) = - 3 \cos (0 + π)$

Step 6.3.2.2

Add $0$ and $π$ .

$f (0) = - 3 \cos (π)$

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.

$f (0) = - 3 (- \cos (0))$

Step 6.3.2.4

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

$f (0) = - 3 (- 1 \cdot 1)$

Step 6.3.2.5

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

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Step 6.3.2.5.1

Multiply $- 1$ by $1$ .

$f (0) = - 3 \cdot - 1$

Step 6.3.2.5.2

Multiply $- 3$ by $- 1$ .

$f (0) = 3$

Step 6.3.2.6

The final answer is $3$ .

$3$

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $6$ .

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Step 6.4.2.1.1

Factor $6$ out of $12$ .

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

Step 6.4.2.1.2

Cancel the common factor.

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

Step 6.4.2.1.3

Rewrite the expression.

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

Step 6.4.2.2

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

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

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.4.2.3.2

Combine the numerators over the common denominator.

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

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Move $2$ to the left of $π$ .

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

Step 6.4.2.4.2

Add $π$ and $2 π$ .

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

Step 6.4.2.5

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

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

Step 6.4.2.6

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

$f (\frac{π}{12}) = - 3 \cdot 0$

Step 6.4.2.7

Multiply $- 3$ by $0$ .

$f (\frac{π}{12}) = 0$

Step 6.4.2.8

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{π}{6}$ .

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Step 6.5.1

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

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of $6$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Rewrite the expression.

$f (\frac{π}{6}) = - 3 \cos (π + π)$

Step 6.5.2.2

Add $π$ and $π$ .

$f (\frac{π}{6}) = - 3 \cos (2 π)$

Step 6.5.2.3

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

$f (\frac{π}{6}) = - 3 \cos (0)$

Step 6.5.2.4

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

$f (\frac{π}{6}) = - 3 \cdot 1$

Step 6.5.2.5

Multiply $- 3$ by $1$ .

$f (\frac{π}{6}) = - 3$

Step 6.5.2.6

The final answer is $- 3$ .

$- 3$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $3$

Period: $\frac{π}{3}$

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

Vertical Shift: None

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

Step 8