Trigonometry Examples

y = 2 cos (4 x - \frac{π}{4})

$y = 2 \cos (4 x - \frac{π}{4})$

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 = 2

$a = 2$

b = 4

$b = 4$

c = \frac{π}{4}

$c = \frac{π}{4}$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

2

$2$

Step 3

Find the period of

2 cos (4 x - \frac{π}{4})

$2 \cos (4 x - \frac{π}{4})$ .

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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

4

$4$ in the formula for period.

\frac{2 π}{| 4 |}

$\frac{2 π}{| 4 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

4

$4$ is

4

$4$ .

\frac{2 π}{4}

$\frac{2 π}{4}$

Step 3.4

Cancel the common factor of

2

$2$ and

4

$4$ .

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Step 3.4.1

Factor

2

$2$ out of

2 π

$2 π$ .

\frac{2 (π)}{4}

$\frac{2 (π)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor

2

$2$ out of

4

$4$ .

\frac{2 π}{2 \cdot 2}

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{2}$

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:

$\frac{\frac{π}{4}}{4}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$\frac{π}{4} \cdot \frac{1}{4}$

Step 4.4

Multiply

$\frac{π}{4} \cdot \frac{1}{4}$ .

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Step 4.4.1

Multiply

$\frac{π}{4}$ by

$\frac{1}{4}$ .

Phase Shift:

$\frac{π}{4 \cdot 4}$

Step 4.4.2

Multiply

$4$ by

$4$ .

Phase Shift:

$\frac{π}{16}$

Phase Shift:

$\frac{π}{16}$

Phase Shift:

$\frac{π}{16}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$2$

Period:

$\frac{π}{2}$

Phase Shift:

$\frac{π}{16}$ (

$\frac{π}{16}$ to the right)

Vertical Shift: None

Step 6

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