Trigonometry Examples

$y = - 5 \cos (\frac{x}{13} + \frac{π}{6})$

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.

$a = - 5$

$b = \frac{1}{13}$

$c = - \frac{π}{6}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $5$

Step 3

Find the period of $- 5 \cos (\frac{x}{13} + \frac{π}{6})$ .

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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 $\frac{1}{13}$ in the formula for period.

$\frac{2 π}{| \frac{1}{13} |}$

Step 3.3

$\frac{1}{13}$ is approximately $0.\overline{076923}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{13}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 13$

Step 3.5

Multiply $13$ by $2$ .

$26 π$

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: $\frac{- \frac{π}{6}}{\frac{1}{13}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot 13$

Step 4.4

Multiply $- \frac{π}{6} \cdot 13$ .

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Step 4.4.1

Multiply $13$ by $- 1$ .

Phase Shift: $- 13 \frac{π}{6}$

Step 4.4.2

Combine $- 13$ and $\frac{π}{6}$ .

Phase Shift: $\frac{- 13 π}{6}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{13 π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $5$

Period: $26 π$

Phase Shift: $- \frac{13 π}{6}$ ( $\frac{13 π}{6}$ to the left)

Vertical Shift: None

Step 6