Trigonometry Examples

y = - 2 cos (x - π) - 2

$y = - 2 \cos (x - π) - 2$

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 = - 2

$a = - 2$

b = 1

$b = 1$

c = π

$c = π$

d = - 2

$d = - 2$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

2

$2$

Step 3

Find the period using the formula

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

Tap for more steps...

Step 3.1

Find the period of

- 2 cos (x - π)

$- 2 \cos (x - π)$ .

Tap for more steps...

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

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.1.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.1.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 3.2

Find the period of

- 2

$- 2$ .

Tap for more steps...

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

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.2.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.2.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 3.3

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

2 π

$2 π$

2 π

$2 π$

Step 4

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

Tap for more steps...

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{π}{1}

$\frac{π}{1}$

Step 4.3

Divide

π

$π$ by

1

$1$ .

Phase Shift:

π

$π$

Phase Shift:

π

$π$

Step 5

List the properties of the trigonometric function.

Amplitude:

2

$2$

Period:

2 π

$2 π$

Phase Shift:

π

$π$ (

π

$π$ to the right)

Vertical Shift:

- 2

$- 2$

Step 6