Trigonometry Examples

$f (x) = \cos (3 π x - π)$

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.

$a = 1$

$b = 3 π$

$c = π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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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 $3 π$ in the formula for period.

$\frac{2 π}{| 3 π |}$

Step 3.3

$3 π$ is approximately $9.42477796$ which is positive so remove the absolute value

$\frac{2 π}{3 π}$

Step 3.4

Cancel the common factor of $π$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{3 π}$

Step 3.4.2

Rewrite the expression.

$\frac{2}{3}$

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: $\frac{π}{3 π}$

Step 4.3

Cancel the common factor of $π$ .

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Step 4.3.1

Cancel the common factor.

Phase Shift: $\frac{π}{3 π}$

Step 4.3.2

Rewrite the expression.

Phase Shift: $\frac{1}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2}{3}$

Phase Shift: $\frac{1}{3}$ ( $\frac{1}{3}$ to the right)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = \frac{1}{3}$ .

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Step 6.1.1

Replace the variable $x$ with $\frac{1}{3}$ in the expression.

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1

Factor $3$ out of $3 π$ .

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

Step 6.1.2.1.2

Cancel the common factor.

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

Step 6.1.2.1.3

Rewrite the expression.

$f (\frac{1}{3}) = \cos (π - π)$

Step 6.1.2.2

Subtract $π$ from $π$ .

$f (\frac{1}{3}) = \cos (0)$

Step 6.1.2.3

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

$f (\frac{1}{3}) = 1$

Step 6.1.2.4

The final answer is $1$ .

$1$

Step 6.2

Find the point at $x = \frac{1}{2}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Multiply $3 π (\frac{1}{2})$ .

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Step 6.2.2.1.1

Combine $\frac{1}{2}$ and $3$ .

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

Step 6.2.2.1.2

Combine $\frac{3}{2}$ and $π$ .

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Combine fractions.

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Step 6.2.2.3.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.2.2.3.2

Combine the numerators over the common denominator.

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

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Multiply $2$ by $- 1$ .

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

Step 6.2.2.4.2

Subtract $2 π$ from $3 π$ .

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

Step 6.2.2.5

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

$f (\frac{1}{2}) = 0$

Step 6.2.2.6

The final answer is $0$ .

$0$

Step 6.3

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

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Step 6.3.1

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

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

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

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.1

Factor $3$ out of $3 π$ .

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

Step 6.3.2.1.1.2

Cancel the common factor.

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

Step 6.3.2.1.1.3

Rewrite the expression.

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

Step 6.3.2.1.2

Move $2$ to the left of $π$ .

$f (\frac{2}{3}) = \cos (2 π - π)$

Step 6.3.2.2

Subtract $π$ from $2 π$ .

$f (\frac{2}{3}) = \cos (π)$

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.

$f (\frac{2}{3}) = - \cos (0)$

Step 6.3.2.4

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

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

Step 6.3.2.5

Multiply $- 1$ by $1$ .

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

Step 6.3.2.6

The final answer is $- 1$ .

$- 1$

Step 6.4

Find the point at $x = \frac{5}{6}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{5}{6}$ in the expression.

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

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 $3$ .

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Step 6.4.2.1.1.1

Factor $3$ out of $3 π$ .

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

Step 6.4.2.1.1.2

Factor $3$ out of $6$ .

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

Step 6.4.2.1.1.3

Cancel the common factor.

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

Step 6.4.2.1.1.4

Rewrite the expression.

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

Step 6.4.2.1.2

Combine $π$ and $\frac{5}{2}$ .

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

Step 6.4.2.1.3

Move $5$ to the left of $π$ .

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

Step 6.4.2.2

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

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

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.4.2.3.2

Combine the numerators over the common denominator.

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

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply $2$ by $- 1$ .

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

Step 6.4.2.4.2

Subtract $2 π$ from $5 π$ .

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

Step 6.4.2.5

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

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

Step 6.4.2.6

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

$f (\frac{5}{6}) = 0$

Step 6.4.2.7

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = 1$ .

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Step 6.5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \cos (3 π (1) - π)$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Multiply $3$ by $1$ .

$f (1) = \cos (3 π - π)$

Step 6.5.2.2

Subtract $π$ from $3 π$ .

$f (1) = \cos (2 π)$

Step 6.5.2.3

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

$f (1) = \cos (0)$

Step 6.5.2.4

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

$f (1) = 1$

Step 6.5.2.5

The final answer is $1$ .

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{1}{3} & 1 \\ \frac{1}{2} & 0 \\ \frac{2}{3} & - 1 \\ \frac{5}{6} & 0 \\ 1 & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $\frac{2}{3}$

Phase Shift: $\frac{1}{3}$ ( $\frac{1}{3}$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{1}{3} & 1 \\ \frac{1}{2} & 0 \\ \frac{2}{3} & - 1 \\ \frac{5}{6} & 0 \\ 1 & 1 \end{array}$

Step 8