Trigonometry Examples

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

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

Step 1

Use the form

a cot (b x - c) + d

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

a = 2

$a = 2$

b = \frac{1}{3}

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

c = - \frac{π}{6}

$c = - \frac{π}{6}$

d = 2

$d = 2$

Step 2

Since the graph of the function

c o t

$c o t$ 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 |}

$\frac{π}{| b |}$ .

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Step 3.1

Find the period of

2 cot (\frac{x}{3} + \frac{π}{6})

$2 \cot (\frac{x}{3} + \frac{π}{6})$ .

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Step 3.1.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 3.1.2

Replace

b

$b$ with

\frac{1}{3}

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

\frac{π}{∣ ∣ \frac{1}{3} ∣ ∣}

$\frac{π}{| \frac{1}{3} |}$

Step 3.1.3

\frac{1}{3}

$\frac{1}{3}$ is approximately

0. ¯ 3

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

\frac{π}{\frac{1}{3}}

$\frac{π}{\frac{1}{3}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

π \cdot 3

$π \cdot 3$

Step 3.1.5

Move

3

$3$ to the left of

π

$π$ .

3 π

$3 π$

3 π

$3 π$

Step 3.2

Find the period of

2

$2$ .

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Step 3.2.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 3.2.2

Replace

b

$b$ with

\frac{1}{3}

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

\frac{π}{∣ ∣ \frac{1}{3} ∣ ∣}

$\frac{π}{| \frac{1}{3} |}$

Step 3.2.3

\frac{1}{3}

$\frac{1}{3}$ is approximately

0. ¯ 3

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

\frac{π}{\frac{1}{3}}

$\frac{π}{\frac{1}{3}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

π \cdot 3

$π \cdot 3$

Step 3.2.5

Move

3

$3$ to the left of

π

$π$ .

3 π

$3 π$

3 π

$3 π$

Step 3.3

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

3 π

$3 π$

3 π

$3 π$

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{- \frac{π}{6}}{\frac{1}{3}}

$\frac{- \frac{π}{6}}{\frac{1}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

- \frac{π}{6} \cdot 3

$- \frac{π}{6} \cdot 3$

Step 4.4

Cancel the common factor of

3

$3$ .

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Step 4.4.1

Move the leading negative in

- \frac{π}{6}

$- \frac{π}{6}$ into the numerator.

Phase Shift:

\frac{- π}{6} \cdot 3

$\frac{- π}{6} \cdot 3$

Step 4.4.2

Factor

3

$3$ out of

6

$6$ .

Phase Shift:

\frac{- π}{3 (2)} \cdot 3

$\frac{- π}{3 (2)} \cdot 3$

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:

$\frac{- π}{2}$

Phase Shift:

$\frac{- π}{2}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift:

$- \frac{π}{2}$

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