Trigonometry Examples

f (x) = - \frac{1}{2} cos (4 (x + \frac{π}{4})) + 1

$f (x) = - \frac{1}{2} \cos (4 (x + \frac{π}{4})) + 1$

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}{2}

$a = - \frac{1}{2}$

b = 4

$b = 4$

c = - π

$c = - π$

d = 1

$d = 1$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

\frac{1}{2}

$\frac{1}{2}$

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

- \frac{cos (4 x + π)}{2}

$- \frac{\cos (4 x + π)}{2}$ .

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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

4

$4$ in the formula for period.

\frac{2 π}{| 4 |}

$\frac{2 π}{| 4 |}$

Step 3.1.3

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 3.1.4

Cancel the common factor of

2

$2$ and

4

$4$ .

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Step 3.1.4.1

Factor

2

$2$ out of

2 π

$2 π$ .

\frac{2 (π)}{4}

$\frac{2 (π)}{4}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor

2

$2$ out of

4

$4$ .

\frac{2 π}{2 \cdot 2}

$\frac{2 π}{2 \cdot 2}$

Step 3.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.1.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 3.2

Find the period of

$1$ .

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Step 3.2.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.2

Replace

$b$ with

$4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 3.2.3

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

$0$ and

$4$ is

$4$ .

$\frac{2 π}{4}$

Step 3.2.4

Cancel the common factor of

$2$ and

$4$ .

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Step 3.2.4.1

Factor

$2$ out of

$2 π$ .

$\frac{2 (π)}{4}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor

$2$ out of

$4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 3.3

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

$\frac{π}{2}$

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:

$\frac{- π}{4}$

Step 4.3

Move the negative in front of the fraction.

Phase Shift:

$- \frac{π}{4}$

Phase Shift:

$- \frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$\frac{1}{2}$

Period:

$\frac{π}{2}$

Phase Shift:

$- \frac{π}{4}$ (

$\frac{π}{4}$ to the left)

Vertical Shift:

$1$

Step 6