Trigonometry Examples

y = 4 cos (\frac{6 π x}{7} - \frac{1}{2})

$y = 4 \cos (\frac{6 π x}{7} - \frac{1}{2})$

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 = 4

$a = 4$

b = \frac{6 π}{7}

$b = \frac{6 π}{7}$

c = \frac{1}{2}

$c = \frac{1}{2}$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

4

$4$

Step 3

Find the period of

4 cos (\frac{6 π x}{7} - \frac{1}{2})

$4 \cos (\frac{6 π x}{7} - \frac{1}{2})$ .

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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{6 π}{7}

$\frac{6 π}{7}$ in the formula for period.

\frac{2 π}{∣ ∣ \frac{6 π}{7} ∣ ∣}

$\frac{2 π}{| \frac{6 π}{7} |}$

Step 3.3

\frac{6 π}{7}

$\frac{6 π}{7}$ is approximately

2.6927937

$2.6927937$ which is positive so remove the absolute value

\frac{2 π}{\frac{6 π}{7}}

$\frac{2 π}{\frac{6 π}{7}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

2 π \frac{7}{6 π}

$2 π \frac{7}{6 π}$

Step 3.5

Cancel the common factor of

2 π

$2 π$ .

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Step 3.5.1

Factor

2 π

$2 π$ out of

6 π

$6 π$ .

2 π \frac{7}{2 π (3)}

$2 π \frac{7}{2 π (3)}$

Step 3.5.2

Cancel the common factor.

$2 π \frac{7}{2 π \cdot 3}$

Step 3.5.3

Rewrite the expression.

$\frac{7}{3}$

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:

$\frac{\frac{1}{2}}{\frac{6 π}{7}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$\frac{1}{2} \cdot \frac{7}{6 π}$

Step 4.4

Multiply

$\frac{1}{2} \cdot \frac{7}{6 π}$ .

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Step 4.4.1

Multiply

$\frac{1}{2}$ by

$\frac{7}{6 π}$ .

Phase Shift:

$\frac{7}{2 (6 π)}$

Step 4.4.2

Multiply

$6$ by

$2$ .

Phase Shift:

$\frac{7}{12 π}$

Phase Shift:

$\frac{7}{12 π}$

Phase Shift:

$\frac{7}{12 π}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$4$

Period:

$\frac{7}{3}$

Phase Shift:

$\frac{7}{12 π}$ (

$\frac{7}{12 π}$ to the right)

Vertical Shift: None

Step 6