Trigonometry Examples

- 5 \cos (\frac{π}{8} x) + 3

$- 5 \cos (\frac{π}{8} x) + 3$

Step 1

Use the form

a \cos (b x - c) + d

$a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a = - 5

$a = - 5$

b = \frac{π}{8}

$b = \frac{π}{8}$

c = 0

$c = 0$

d = 3

$d = 3$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

5

$5$

Step 3

Find the period using the formula

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of

- 5 \cos (\frac{π x}{8})

$- 5 \cos (\frac{π x}{8})$ .

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Step 3.1.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.1.2

Replace

b

$b$ with

\frac{π}{8}

$\frac{π}{8}$ in the formula for period.

\frac{2 π}{| \frac{π}{8} |}

$\frac{2 π}{| \frac{π}{8} |}$

Step 3.1.3

\frac{π}{8}

$\frac{π}{8}$ is approximately

0.39269908

$0.39269908$ which is positive so remove the absolute value

\frac{2 π}{\frac{π}{8}}

$\frac{2 π}{\frac{π}{8}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

2 π \frac{8}{π}

$2 π \frac{8}{π}$

Step 3.1.5

Cancel the common factor of

π

$π$ .

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Step 3.1.5.1

Factor

π

$π$ out of

2 π

$2 π$ .

π \cdot 2 \frac{8}{π}

$π \cdot 2 \frac{8}{π}$

Step 3.1.5.2

Cancel the common factor.

π \cdot 2 \frac{8}{π}

$π \cdot 2 \frac{8}{π}$

Step 3.1.5.3

Rewrite the expression.

2 \cdot 8

$2 \cdot 8$

2 \cdot 8

$2 \cdot 8$

Step 3.1.6

Multiply

2

$2$ by

8

$8$ .

16

$16$

16

$16$

Step 3.2

Find the period of

3

$3$ .

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Step 3.2.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2.2

Replace

b

$b$ with

\frac{π}{8}

$\frac{π}{8}$ in the formula for period.

\frac{2 π}{| \frac{π}{8} |}

$\frac{2 π}{| \frac{π}{8} |}$

Step 3.2.3

\frac{π}{8}

$\frac{π}{8}$ is approximately

0.39269908

$0.39269908$ which is positive so remove the absolute value

\frac{2 π}{\frac{π}{8}}

$\frac{2 π}{\frac{π}{8}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

2 π \frac{8}{π}

$2 π \frac{8}{π}$

Step 3.2.5

Cancel the common factor of

π

$π$ .

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Step 3.2.5.1

Factor

π

$π$ out of

2 π

$2 π$ .

π \cdot 2 \frac{8}{π}

$π \cdot 2 \frac{8}{π}$

Step 3.2.5.2

Cancel the common factor.

π \cdot 2 \frac{8}{π}

$π \cdot 2 \frac{8}{π}$

Step 3.2.5.3

Rewrite the expression.

2 \cdot 8

$2 \cdot 8$

2 \cdot 8

$2 \cdot 8$

Step 3.2.6

Multiply

2

$2$ by

8

$8$ .

16

$16$

16

$16$

Step 3.3

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

16

$16$

16

$16$

Step 4

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 4.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{0}{\frac{π}{8}}

$\frac{0}{\frac{π}{8}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

0 (\frac{8}{π})

$0 (\frac{8}{π})$

Step 4.4

Multiply

0

$0$ by

\frac{8}{π}

$\frac{8}{π}$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

5

$5$

Period:

16

$16$

Phase Shift: None

Vertical Shift:

3

$3$

Step 6