Trigonometry Examples

y = \frac{1}{2} \cos (2 x - π) + 2

$y = \frac{1}{2} \cos (2 x - π) + 2$

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 = \frac{1}{2}

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

b = 2

$b = 2$

c = π

$c = π$

d = 2

$d = 2$

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 |}

$\frac{2 π}{| b |}$ .

Tap for more steps...

Step 3.1

Find the period of

\frac{\cos (2 x - π)}{2}

$\frac{\cos (2 x - π)}{2}$ .

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

2

$2$ in the formula for period.

\frac{2 π}{| 2 |}

$\frac{2 π}{| 2 |}$

Step 3.1.3

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

0

$0$ and

2

$2$ is

2

$2$ .

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 3.1.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 3.1.4.1

Cancel the common factor.

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 3.1.4.2

Divide

π

$π$ by

1

$1$ .

π

$π$

π

$π$

π

$π$

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

2

$2$ in the formula for period.

\frac{2 π}{| 2 |}

$\frac{2 π}{| 2 |}$

Step 3.2.3

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

0

$0$ and

2

$2$ is

2

$2$ .

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 3.2.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 3.2.4.1

Cancel the common factor.

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 3.2.4.2

Divide

π

$π$ by

1

$1$ .

π

$π$

π

$π$

π

$π$

Step 3.3

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

π

$π$

π

$π$

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}{2}

$\frac{π}{2}$

Phase Shift:

\frac{π}{2}

$\frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude:

\frac{1}{2}

$\frac{1}{2}$

Period:

π

$π$

Phase Shift:

\frac{π}{2}

$\frac{π}{2}$ (

\frac{π}{2}

$\frac{π}{2}$ to the right)

Vertical Shift:

2

$2$

Step 6