Trigonometry Examples

$y = 3 \cos (2 π x + 4 π)$

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.

$a = 3$

$b = 2 π$

$c = - 4 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $3 \cos (2 π x + 4 π)$ .

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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 $2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 3.3

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 3.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 3.5

Cancel the common factor of $π$ .

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Step 3.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 3.5.2

Rewrite the expression.

$1$

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: $\frac{- 4 π}{2 π}$

Step 4.3

Cancel the common factor of $- 4$ and $2$ .

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Step 4.3.1

Factor $2$ out of $- 4 π$ .

Phase Shift: $\frac{2 (- 2 π)}{2 π}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2 π$ .

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

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{2 (- 2 π)}{2 π}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{- 2 π}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Cancel the common factor.

Phase Shift: $\frac{- 2 π}{π}$

Step 4.4.2

Divide $- 2$ by $1$ .

Phase Shift: $- 2$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $1$

Phase Shift: $- 2$ ( $2$ 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 $x = - 2$ .

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Step 6.1.1

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

$f (- 2) = 3 \cos (2 π (- 2) + 4 π)$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $- 2$ by $2$ .

$f (- 2) = 3 \cos (- 4 π + 4 π)$

Step 6.1.2.2

Add $- 4 π$ and $4 π$ .

$f (- 2) = 3 \cos (0)$

Step 6.1.2.3

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

$f (- 2) = 3 \cdot 1$

Step 6.1.2.4

Multiply $3$ by $1$ .

$f (- 2) = 3$

Step 6.1.2.5

The final answer is $3$ .

$3$

Step 6.2

Find the point at $x = - \frac{7}{4}$ .

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Step 6.2.1

Replace the variable $x$ with $- \frac{7}{4}$ in the expression.

$f (- \frac{7}{4}) = 3 \cos (2 π (- \frac{7}{4}) + 4 π)$

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 $2$ .

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Step 6.2.2.1.1.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$f (- \frac{7}{4}) = 3 \cos (2 π (\frac{- 7}{4}) + 4 π)$

Step 6.2.2.1.1.2

Factor $2$ out of $2 π$ .

$f (- \frac{7}{4}) = 3 \cos (2 (π) (\frac{- 7}{4}) + 4 π)$

Step 6.2.2.1.1.3

Factor $2$ out of $4$ .

$f (- \frac{7}{4}) = 3 \cos (2 (π) (\frac{- 7}{2 (2)}) + 4 π)$

Step 6.2.2.1.1.4

Cancel the common factor.

$f (- \frac{7}{4}) = 3 \cos (2 π (\frac{- 7}{2 \cdot 2}) + 4 π)$

Step 6.2.2.1.1.5

Rewrite the expression.

$f (- \frac{7}{4}) = 3 \cos (π (\frac{- 7}{2}) + 4 π)$

Step 6.2.2.1.2

Combine $π$ and $\frac{- 7}{2}$ .

$f (- \frac{7}{4}) = 3 \cos (\frac{π \cdot - 7}{2} + 4 π)$

Step 6.2.2.1.3

Move $- 7$ to the left of $π$ .

$f (- \frac{7}{4}) = 3 \cos (\frac{- 7 \cdot π}{2} + 4 π)$

Step 6.2.2.1.4

Move the negative in front of the fraction.

$f (- \frac{7}{4}) = 3 \cos (- \frac{7 π}{2} + 4 π)$

Step 6.2.2.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{7}{4}) = 3 \cos (- \frac{7 π}{2} + 4 π \cdot \frac{2}{2})$

Step 6.2.2.3

Combine fractions.

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Step 6.2.2.3.1

Combine $4 π$ and $\frac{2}{2}$ .

$f (- \frac{7}{4}) = 3 \cos (- \frac{7 π}{2} + \frac{4 π \cdot 2}{2})$

Step 6.2.2.3.2

Combine the numerators over the common denominator.

$f (- \frac{7}{4}) = 3 \cos (\frac{- 7 π + 4 π \cdot 2}{2})$

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Multiply $2$ by $4$ .

$f (- \frac{7}{4}) = 3 \cos (\frac{- 7 π + 8 π}{2})$

Step 6.2.2.4.2

Add $- 7 π$ and $8 π$ .

$f (- \frac{7}{4}) = 3 \cos (\frac{π}{2})$

Step 6.2.2.5

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

$f (- \frac{7}{4}) = 3 \cdot 0$

Step 6.2.2.6

Multiply $3$ by $0$ .

$f (- \frac{7}{4}) = 0$

Step 6.2.2.7

The final answer is $0$ .

$0$

Step 6.3

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

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Step 6.3.1

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

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

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 $2$ .

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Step 6.3.2.1.1.1

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

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

Step 6.3.2.1.1.2

Factor $2$ out of $2 π$ .

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

Step 6.3.2.1.1.3

Cancel the common factor.

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

Step 6.3.2.1.1.4

Rewrite the expression.

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

Step 6.3.2.1.2

Move $- 3$ to the left of $π$ .

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

Step 6.3.2.2

Add $- 3 π$ and $4 π$ .

$f (- \frac{3}{2}) = 3 \cos (π)$

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.

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

Step 6.3.2.4

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

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

Step 6.3.2.5

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

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Step 6.3.2.5.1

Multiply $- 1$ by $1$ .

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

Step 6.3.2.5.2

Multiply $3$ by $- 1$ .

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

Step 6.3.2.6

The final answer is $- 3$ .

$- 3$

Step 6.4

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

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Step 6.4.1

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

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

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 $2$ .

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Step 6.4.2.1.1.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

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

Step 6.4.2.1.1.2

Factor $2$ out of $2 π$ .

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

Step 6.4.2.1.1.3

Factor $2$ out of $4$ .

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

Step 6.4.2.1.1.4

Cancel the common factor.

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

Step 6.4.2.1.1.5

Rewrite the expression.

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

Step 6.4.2.1.2

Combine $π$ and $\frac{- 5}{2}$ .

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

Step 6.4.2.1.3

Move $- 5$ to the left of $π$ .

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

Step 6.4.2.1.4

Move the negative in front of the fraction.

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

Step 6.4.2.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $4 π$ and $\frac{2}{2}$ .

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

Step 6.4.2.3.2

Combine the numerators over the common denominator.

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

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply $2$ by $4$ .

$f (- \frac{5}{4}) = 3 \cos (\frac{- 5 π + 8 π}{2})$

Step 6.4.2.4.2

Add $- 5 π$ and $8 π$ .

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

Step 6.4.2.5

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

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

Step 6.4.2.6

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

$f (- \frac{5}{4}) = 3 \cdot 0$

Step 6.4.2.7

Multiply $3$ by $0$ .

$f (- \frac{5}{4}) = 0$

Step 6.4.2.8

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.

$f (- 1) = 3 \cos (2 π (- 1) + 4 π)$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Multiply $- 1$ by $2$ .

$f (- 1) = 3 \cos (- 2 π + 4 π)$

Step 6.5.2.2

Add $- 2 π$ and $4 π$ .

$f (- 1) = 3 \cos (2 π)$

Step 6.5.2.3

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

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

Step 6.5.2.4

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

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

Step 6.5.2.5

Multiply $3$ by $1$ .

$f (- 1) = 3$

Step 6.5.2.6

The final answer is $3$ .

$3$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 2 & 3 \\ - \frac{7}{4} & 0 \\ - \frac{3}{2} & - 3 \\ - \frac{5}{4} & 0 \\ - 1 & 3 \end{array}$

Step 7

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

Amplitude: $3$

Period: $1$

Phase Shift: $- 2$ ( $2$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 2 & 3 \\ - \frac{7}{4} & 0 \\ - \frac{3}{2} & - 3 \\ - \frac{5}{4} & 0 \\ - 1 & 3 \end{array}$

Step 8