Trigonometry Examples

y = 3 \sin (\frac{θ}{4}) - 2

$y = 3 \sin (\frac{θ}{4}) - 2$

Step 1

Use the form

a \sin (b θ - c) + d

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

a = 3

$a = 3$

b = \frac{1}{4}

$b = \frac{1}{4}$

c = 0

$c = 0$

d = - 2

$d = - 2$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

3

$3$

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

3 \sin (\frac{θ}{4})

$3 \sin (\frac{θ}{4})$ .

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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{1}{4}

$\frac{1}{4}$ in the formula for period.

\frac{2 π}{| \frac{1}{4} |}

$\frac{2 π}{| \frac{1}{4} |}$

Step 3.1.3

\frac{1}{4}

$\frac{1}{4}$ is approximately

0.25

$0.25$ which is positive so remove the absolute value

\frac{2 π}{\frac{1}{4}}

$\frac{2 π}{\frac{1}{4}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

2 π \cdot 4

$2 π \cdot 4$

Step 3.1.5

Multiply

4

$4$ by

2

$2$ .

8 π

$8 π$

8 π

$8 π$

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{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2.2

Replace

b

$b$ with

\frac{1}{4}

$\frac{1}{4}$ in the formula for period.

\frac{2 π}{| \frac{1}{4} |}

$\frac{2 π}{| \frac{1}{4} |}$

Step 3.2.3

\frac{1}{4}

$\frac{1}{4}$ is approximately

0.25

$0.25$ which is positive so remove the absolute value

\frac{2 π}{\frac{1}{4}}

$\frac{2 π}{\frac{1}{4}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

2 π \cdot 4

$2 π \cdot 4$

Step 3.2.5

Multiply

4

$4$ by

2

$2$ .

8 π

$8 π$

8 π

$8 π$

Step 3.3

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

8 π

$8 π$

8 π

$8 π$

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{0}{\frac{1}{4}}

$\frac{0}{\frac{1}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

0 \cdot 4

$0 \cdot 4$

Step 4.4

Multiply

0

$0$ by

4

$4$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

3

$3$

Period:

8 π

$8 π$

Phase Shift: None

Vertical Shift:

- 2

$- 2$

Step 6