Trigonometry Examples

$f (x) = 3 \cos (2 x - π) - 1$

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.

$a = 3$

$b = 2$

$c = π$

$d = - 1$

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 $3 \cos (2 x - π)$ .

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.1.3

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

$\frac{2 π}{2}$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.1.4.2

Divide $π$ by $1$ .

$π$

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.2.3

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

$\frac{2 π}{2}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.4.2

Divide $π$ by $1$ .

$π$

Step 3.3

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

$π$

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: $\frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ 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 $x = \frac{π}{2}$ .

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Step 6.1.1

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

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

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

Cancel the common factor of $2$ .

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Step 6.1.2.1.1.1

Cancel the common factor.

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

Step 6.1.2.1.1.2

Rewrite the expression.

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

Step 6.1.2.1.2

Subtract $π$ from $π$ .

$f (\frac{π}{2}) = 3 \cos (0) - 1$

Step 6.1.2.1.3

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

$f (\frac{π}{2}) = 3 \cdot 1 - 1$

Step 6.1.2.1.4

Multiply $3$ by $1$ .

$f (\frac{π}{2}) = 3 - 1$

Step 6.1.2.2

Subtract $1$ from $3$ .

$f (\frac{π}{2}) = 2$

Step 6.1.2.3

The final answer is $2$ .

$2$

Step 6.2

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

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Step 6.2.1

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

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

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

Factor $2$ out of $4$ .

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

Step 6.2.2.1.1.2

Cancel the common factor.

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

Step 6.2.2.1.1.3

Rewrite the expression.

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

Step 6.2.2.1.2

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

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

Step 6.2.2.1.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.2.2.1.4

Combine the numerators over the common denominator.

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

Step 6.2.2.1.5

Simplify the numerator.

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Step 6.2.2.1.5.1

Multiply $2$ by $- 1$ .

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

Step 6.2.2.1.5.2

Subtract $2 π$ from $3 π$ .

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

Step 6.2.2.1.6

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

$f (\frac{3 π}{4}) = 3 \cdot 0 - 1$

Step 6.2.2.1.7

Multiply $3$ by $0$ .

$f (\frac{3 π}{4}) = 0 - 1$

Step 6.2.2.2

Subtract $1$ from $0$ .

$f (\frac{3 π}{4}) = - 1$

Step 6.2.2.3

The final answer is $- 1$ .

$- 1$

Step 6.3

Find the point at $x = π$ .

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Step 6.3.1

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

$f (π) = 3 \cos (2 (π) - π) - 1$

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

Subtract $π$ from $2 π$ .

$f (π) = 3 \cos (π) - 1$

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.

$f (π) = 3 (- \cos (0)) - 1$

Step 6.3.2.1.3

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

$f (π) = 3 (- 1 \cdot 1) - 1$

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 $1$ .

$f (π) = 3 \cdot - 1 - 1$

Step 6.3.2.1.4.2

Multiply $3$ by $- 1$ .

$f (π) = - 3 - 1$

Step 6.3.2.2

Subtract $1$ from $- 3$ .

$f (π) = - 4$

Step 6.3.2.3

The final answer is $- 4$ .

$- 4$

Step 6.4

Find the point at $x = \frac{5 π}{4}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{5 π}{4}$ in the expression.

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

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

Cancel the common factor of $2$ .

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Step 6.4.2.1.1.1

Factor $2$ out of $4$ .

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

Step 6.4.2.1.1.2

Cancel the common factor.

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

Step 6.4.2.1.1.3

Rewrite the expression.

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

Step 6.4.2.1.2

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

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

Step 6.4.2.1.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.4.2.1.4

Combine the numerators over the common denominator.

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

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Multiply $2$ by $- 1$ .

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

Step 6.4.2.1.5.2

Subtract $2 π$ from $5 π$ .

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

Step 6.4.2.1.6

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

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

Step 6.4.2.1.7

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

$f (\frac{5 π}{4}) = 3 \cdot 0 - 1$

Step 6.4.2.1.8

Multiply $3$ by $0$ .

$f (\frac{5 π}{4}) = 0 - 1$

Step 6.4.2.2

Subtract $1$ from $0$ .

$f (\frac{5 π}{4}) = - 1$

Step 6.4.2.3

The final answer is $- 1$ .

$- 1$

Step 6.5

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

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Step 6.5.1

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

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

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

Cancel the common factor of $2$ .

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Step 6.5.2.1.1.1

Cancel the common factor.

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

Step 6.5.2.1.1.2

Rewrite the expression.

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

Step 6.5.2.1.2

Subtract $π$ from $3 π$ .

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

Step 6.5.2.1.3

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

$f (\frac{3 π}{2}) = 3 \cos (0) - 1$

Step 6.5.2.1.4

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

$f (\frac{3 π}{2}) = 3 \cdot 1 - 1$

Step 6.5.2.1.5

Multiply $3$ by $1$ .

$f (\frac{3 π}{2}) = 3 - 1$

Step 6.5.2.2

Subtract $1$ from $3$ .

$f (\frac{3 π}{2}) = 2$

Step 6.5.2.3

The final answer is $2$ .

$2$

Step 6.6

List the points in a table.

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

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: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $- 1$

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

Step 8