Trigonometry Examples

y = \frac{1}{4} \cos (\frac{2 x}{7} + \frac{1}{3})

$y = \frac{1}{4} \cos (\frac{2 x}{7} + \frac{1}{3})$

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 = \frac{1}{4}

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

b = \frac{2}{7}

$b = \frac{2}{7}$

c = - \frac{1}{3}

$c = - \frac{1}{3}$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

\frac{1}{4}

$\frac{1}{4}$

Step 3

Find the period of

\frac{\cos (\frac{2 x}{7} + \frac{1}{3})}{4}

$\frac{\cos (\frac{2 x}{7} + \frac{1}{3})}{4}$ .

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Step 3.1

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

Replace

b

$b$ with

\frac{2}{7}

$\frac{2}{7}$ in the formula for period.

\frac{2 π}{| \frac{2}{7} |}

$\frac{2 π}{| \frac{2}{7} |}$

Step 3.3

\frac{2}{7}

$\frac{2}{7}$ is approximately

0.\overline{285714}

$0.\overline{285714}$ which is positive so remove the absolute value

\frac{2 π}{\frac{2}{7}}

$\frac{2 π}{\frac{2}{7}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

2 π \frac{7}{2}

$2 π \frac{7}{2}$

Step 3.5

Cancel the common factor of

2

$2$ .

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Step 3.5.1

Factor

2

$2$ out of

2 π

$2 π$ .

2 (π) \frac{7}{2}

$2 (π) \frac{7}{2}$

Step 3.5.2

Cancel the common factor.

2 π \frac{7}{2}

$2 π \frac{7}{2}$

Step 3.5.3

Rewrite the expression.

π \cdot 7

$π \cdot 7$

π \cdot 7

$π \cdot 7$

Step 3.6

Move

7

$7$ to the left of

π

$π$ .

7 π

$7 π$

7 π

$7 π$

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{1}{3}}{\frac{2}{7}}

$\frac{- \frac{1}{3}}{\frac{2}{7}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

- \frac{1}{3} \cdot \frac{7}{2}

$- \frac{1}{3} \cdot \frac{7}{2}$

Step 4.4

Multiply

- \frac{1}{3} \cdot \frac{7}{2}

$- \frac{1}{3} \cdot \frac{7}{2}$ .

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Step 4.4.1

Multiply

\frac{7}{2}

$\frac{7}{2}$ by

\frac{1}{3}

$\frac{1}{3}$ .

Phase Shift:

- \frac{7}{2 \cdot 3}

$- \frac{7}{2 \cdot 3}$

Step 4.4.2

Multiply

2

$2$ by

3

$3$ .

Phase Shift:

- \frac{7}{6}

$- \frac{7}{6}$

Phase Shift:

- \frac{7}{6}

$- \frac{7}{6}$

Phase Shift:

- \frac{7}{6}

$- \frac{7}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude:

\frac{1}{4}

$\frac{1}{4}$

Period:

7 π

$7 π$

Phase Shift:

- \frac{7}{6}

$- \frac{7}{6}$ (

\frac{7}{6}

$\frac{7}{6}$ to the left)

Vertical Shift: None

Step 6