Trigonometry Examples

$f (x) = - 9 \cos (\frac{π x}{10})$

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.

$a = - 9$

$b = \frac{π}{10}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $9$

Step 3

Find the period of $- 9 \cos (\frac{π x}{10})$ .

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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 $\frac{π}{10}$ in the formula for period.

$\frac{2 π}{| \frac{π}{10} |}$

Step 3.3

$\frac{π}{10}$ is approximately $0.31415926$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{10}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{10}{π}$

Step 3.5

Cancel the common factor of $π$ .

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Step 3.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{10}{π}$

Step 3.5.2

Cancel the common factor.

$π \cdot 2 \frac{10}{π}$

Step 3.5.3

Rewrite the expression.

$2 \cdot 10$

Step 3.6

Multiply $2$ by $10$ .

$20$

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: $\frac{0}{\frac{π}{10}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{10}{π})$

Step 4.4

Multiply $0$ by $\frac{10}{π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $9$

Period: $20$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

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

$f (0) = - 9 \cos (\frac{π (0)}{10})$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Cancel the common factor of $0$ and $10$ .

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Step 6.1.2.1.1

Factor $10$ out of $π (0)$ .

$f (0) = - 9 \cos (\frac{10 (π \cdot (0))}{10})$

Step 6.1.2.1.2

Cancel the common factors.

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Step 6.1.2.1.2.1

Factor $10$ out of $10$ .

$f (0) = - 9 \cos (\frac{10 (π \cdot (0))}{10 (1)})$

Step 6.1.2.1.2.2

Cancel the common factor.

$f (0) = - 9 \cos (\frac{10 (π \cdot (0))}{10 \cdot 1})$

Step 6.1.2.1.2.3

Rewrite the expression.

$f (0) = - 9 \cos (\frac{π \cdot (0)}{1})$

Step 6.1.2.1.2.4

Divide $π \cdot (0)$ by $1$ .

$f (0) = - 9 \cos (π \cdot (0))$

Step 6.1.2.2

Multiply $π$ by $0$ .

$f (0) = - 9 \cos (0)$

Step 6.1.2.3

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

$f (0) = - 9 \cdot 1$

Step 6.1.2.4

Multiply $- 9$ by $1$ .

$f (0) = - 9$

Step 6.1.2.5

The final answer is $- 9$ .

$- 9$

Step 6.2

Find the point at $x = 5$ .

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Step 6.2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - 9 \cos (\frac{π (5)}{10})$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Cancel the common factor of $5$ and $10$ .

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Step 6.2.2.1.1

Factor $5$ out of $π (5)$ .

$f (5) = - 9 \cos (\frac{5 π}{10})$

Step 6.2.2.1.2

Cancel the common factors.

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Step 6.2.2.1.2.1

Factor $5$ out of $10$ .

$f (5) = - 9 \cos (\frac{5 π}{5 \cdot 2})$

Step 6.2.2.1.2.2

Cancel the common factor.

$f (5) = - 9 \cos (\frac{5 π}{5 \cdot 2})$

Step 6.2.2.1.2.3

Rewrite the expression.

$f (5) = - 9 \cos (\frac{π}{2})$

Step 6.2.2.2

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

$f (5) = - 9 \cdot 0$

Step 6.2.2.3

Multiply $- 9$ by $0$ .

$f (5) = 0$

Step 6.2.2.4

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = 10$ .

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Step 6.3.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = - 9 \cos (\frac{π (10)}{10})$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Cancel the common factor of $10$ .

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Step 6.3.2.1.1

Cancel the common factor.

$f (10) = - 9 \cos (\frac{π \cdot 10}{10})$

Step 6.3.2.1.2

Divide $π$ by $1$ .

$f (10) = - 9 \cos (π)$

Step 6.3.2.2

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.

$f (10) = - 9 (- \cos (0))$

Step 6.3.2.3

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

$f (10) = - 9 (- 1 \cdot 1)$

Step 6.3.2.4

Multiply $- 9 (- 1 \cdot 1)$ .

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Step 6.3.2.4.1

Multiply $- 1$ by $1$ .

$f (10) = - 9 \cdot - 1$

Step 6.3.2.4.2

Multiply $- 9$ by $- 1$ .

$f (10) = 9$

Step 6.3.2.5

The final answer is $9$ .

$9$

Step 6.4

Find the point at $x = 15$ .

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Step 6.4.1

Replace the variable $x$ with $15$ in the expression.

$f (15) = - 9 \cos (\frac{π (15)}{10})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $15$ and $10$ .

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Step 6.4.2.1.1

Factor $5$ out of $π (15)$ .

$f (15) = - 9 \cos (\frac{5 (π \cdot (3))}{10})$

Step 6.4.2.1.2

Cancel the common factors.

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Step 6.4.2.1.2.1

Factor $5$ out of $10$ .

$f (15) = - 9 \cos (\frac{5 (π \cdot (3))}{5 (2)})$

Step 6.4.2.1.2.2

Cancel the common factor.

$f (15) = - 9 \cos (\frac{5 (π \cdot (3))}{5 \cdot 2})$

Step 6.4.2.1.2.3

Rewrite the expression.

$f (15) = - 9 \cos (\frac{π \cdot (3)}{2})$

Step 6.4.2.2

Move $3$ to the left of $π$ .

$f (15) = - 9 \cos (\frac{3 \cdot π}{2})$

Step 6.4.2.3

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

$f (15) = - 9 \cos (\frac{π}{2})$

Step 6.4.2.4

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

$f (15) = - 9 \cdot 0$

Step 6.4.2.5

Multiply $- 9$ by $0$ .

$f (15) = 0$

Step 6.4.2.6

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = 20$ .

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Step 6.5.1

Replace the variable $x$ with $20$ in the expression.

$f (20) = - 9 \cos (\frac{π (20)}{10})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of $20$ and $10$ .

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Step 6.5.2.1.1

Factor $10$ out of $π (20)$ .

$f (20) = - 9 \cos (\frac{10 (π \cdot (2))}{10})$

Step 6.5.2.1.2

Cancel the common factors.

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Step 6.5.2.1.2.1

Factor $10$ out of $10$ .

$f (20) = - 9 \cos (\frac{10 (π \cdot (2))}{10 (1)})$

Step 6.5.2.1.2.2

Cancel the common factor.

$f (20) = - 9 \cos (\frac{10 (π \cdot (2))}{10 \cdot 1})$

Step 6.5.2.1.2.3

Rewrite the expression.

$f (20) = - 9 \cos (\frac{π \cdot (2)}{1})$

Step 6.5.2.1.2.4

Divide $π \cdot (2)$ by $1$ .

$f (20) = - 9 \cos (π \cdot (2))$

Step 6.5.2.2

Move $2$ to the left of $π$ .

$f (20) = - 9 \cos (2 \cdot π)$

Step 6.5.2.3

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

$f (20) = - 9 \cos (0)$

Step 6.5.2.4

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

$f (20) = - 9 \cdot 1$

Step 6.5.2.5

Multiply $- 9$ by $1$ .

$f (20) = - 9$

Step 6.5.2.6

The final answer is $- 9$ .

$- 9$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & - 9 \\ 5 & 0 \\ 10 & 9 \\ 15 & 0 \\ 20 & - 9 \end{array}$

Step 7

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

Amplitude: $9$

Period: $20$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & - 9 \\ 5 & 0 \\ 10 & 9 \\ 15 & 0 \\ 20 & - 9 \end{array}$

Step 8