Trigonometry Examples

$y = 3 \cos (x + 3 π) + 2$

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.

$a = 3$

$b = 1$

$c = - 3 π$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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Step 3.1

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.1.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2

Find the period of $2$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.2.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.3

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

$2 π$

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: $\frac{- 3 π}{1}$

Step 4.3

Divide $- 3 π$ by $1$ .

Phase Shift: $- 3 π$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $2 π$

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

Vertical Shift: $2$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - 3 π$ .

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Step 6.1.1

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

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

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

Add $- 3 π$ and $3 π$ .

$f (- 3 π) = 3 \cos (0) + 2$

Step 6.1.2.1.2

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

$f (- 3 π) = 3 \cdot 1 + 2$

Step 6.1.2.1.3

Multiply $3$ by $1$ .

$f (- 3 π) = 3 + 2$

Step 6.1.2.2

Add $3$ and $2$ .

$f (- 3 π) = 5$

Step 6.1.2.3

The final answer is $5$ .

$5$

Step 6.2

Find the point at $x = - \frac{5 π}{2}$ .

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Step 6.2.1

Replace the variable $x$ with $- \frac{5 π}{2}$ in the expression.

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

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

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

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

Step 6.2.2.1.2

Combine $3 π$ and $\frac{2}{2}$ .

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

Step 6.2.2.1.3

Combine the numerators over the common denominator.

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

Step 6.2.2.1.4

Simplify the numerator.

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Step 6.2.2.1.4.1

Multiply $2$ by $3$ .

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

Step 6.2.2.1.4.2

Add $- 5 π$ and $6 π$ .

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

Step 6.2.2.1.5

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

$f (- \frac{5 π}{2}) = 3 \cdot 0 + 2$

Step 6.2.2.1.6

Multiply $3$ by $0$ .

$f (- \frac{5 π}{2}) = 0 + 2$

Step 6.2.2.2

Add $0$ and $2$ .

$f (- \frac{5 π}{2}) = 2$

Step 6.2.2.3

The final answer is $2$ .

$2$

Step 6.3

Find the point at $x = - 2 π$ .

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Step 6.3.1

Replace the variable $x$ with $- 2 π$ in the expression.

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

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

Add $- 2 π$ and $3 π$ .

$f (- 2 π) = 3 \cos (π) + 2$

Step 6.3.2.1.2

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.

$f (- 2 π) = 3 (- \cos (0)) + 2$

Step 6.3.2.1.3

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

$f (- 2 π) = 3 (- 1 \cdot 1) + 2$

Step 6.3.2.1.4

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

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Step 6.3.2.1.4.1

Multiply $- 1$ by $1$ .

$f (- 2 π) = 3 \cdot - 1 + 2$

Step 6.3.2.1.4.2

Multiply $3$ by $- 1$ .

$f (- 2 π) = - 3 + 2$

Step 6.3.2.2

Add $- 3$ and $2$ .

$f (- 2 π) = - 1$

Step 6.3.2.3

The final answer is $- 1$ .

$- 1$

Step 6.4

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

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Step 6.4.1

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

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

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

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

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

Step 6.4.2.1.2

Combine $3 π$ and $\frac{2}{2}$ .

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

Step 6.4.2.1.3

Combine the numerators over the common denominator.

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

Step 6.4.2.1.4

Simplify the numerator.

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Step 6.4.2.1.4.1

Multiply $2$ by $3$ .

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

Step 6.4.2.1.4.2

Add $- 3 π$ and $6 π$ .

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

Step 6.4.2.1.5

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

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

Step 6.4.2.1.6

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

$f (- \frac{3 π}{2}) = 3 \cdot 0 + 2$

Step 6.4.2.1.7

Multiply $3$ by $0$ .

$f (- \frac{3 π}{2}) = 0 + 2$

Step 6.4.2.2

Add $0$ and $2$ .

$f (- \frac{3 π}{2}) = 2$

Step 6.4.2.3

The final answer is $2$ .

$2$

Step 6.5

Find the point at $x = - π$ .

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Step 6.5.1

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

$f (- π) = 3 \cos ((- π) + 3 π) + 2$

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

Add $- π$ and $3 π$ .

$f (- π) = 3 \cos (2 π) + 2$

Step 6.5.2.1.2

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

$f (- π) = 3 \cos (0) + 2$

Step 6.5.2.1.3

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

$f (- π) = 3 \cdot 1 + 2$

Step 6.5.2.1.4

Multiply $3$ by $1$ .

$f (- π) = 3 + 2$

Step 6.5.2.2

Add $3$ and $2$ .

$f (- π) = 5$

Step 6.5.2.3

The final answer is $5$ .

$5$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $3$

Period: $2 π$

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

Vertical Shift: $2$

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

Step 8