Trigonometry Examples

$y = 2 \cot (\frac{1}{3} x + \frac{π}{6}) + 2$

Step 1

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

$a = 2$

$b = \frac{1}{3}$

$c = - \frac{π}{6}$

$d = 2$

Step 2

Since the graph of the function $c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 3

Find the period using the formula $\frac{π}{| b |}$ .

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Step 3.1

Find the period of $2 \cot (\frac{x}{3} + \frac{π}{6})$ .

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Step 3.1.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{π}{| \frac{1}{3} |}$

Step 3.1.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{3}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 3$

Step 3.1.5

Move $3$ to the left of $π$ .

$3 π$

Step 3.2

Find the period of $2$ .

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Step 3.2.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{π}{| \frac{1}{3} |}$

Step 3.2.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{3}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 3$

Step 3.2.5

Move $3$ to the left of $π$ .

$3 π$

Step 3.3

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

$3 π$

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: $\frac{- \frac{π}{6}}{\frac{1}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot 3$

Step 4.4

Cancel the common factor of $3$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

Phase Shift: $\frac{- π}{6} \cdot 3$

Step 4.4.2

Factor $3$ out of $6$ .

Phase Shift: $\frac{- π}{3 (2)} \cdot 3$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{- π}{3 \cdot 2} \cdot 3$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- π}{2}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $3 π$

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

Vertical Shift: $2$

Step 6