Trigonometry Examples

$y = 2 - \frac{2}{3} \cos (x - \frac{π}{2})$

Step 1

Rewrite the expression as $- \frac{2 \cos (x - \frac{π}{2})}{3} + 2$ .

$- \frac{2 \cos (x - \frac{π}{2})}{3} + 2$

Step 2

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = - \frac{2}{3}$

$b = 1$

$c = \frac{π}{2}$

$d = 2$

Step 3

Find the amplitude $| a |$ .

Amplitude: $\frac{2}{3}$

Step 4

Find the period using the formula $\frac{2 π}{| b |}$ .

Tap for more steps...

Step 4.1

Find the period of $- \frac{2 \cos (x - \frac{π}{2})}{3}$ .

Tap for more steps...

Step 4.1.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.1.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 4.1.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2

Find the period of $2$ .

Tap for more steps...

Step 4.2.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 4.2.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.3

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

$2 π$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

Tap for more steps...

Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{\frac{π}{2}}{1}$

Step 5.3

Divide $\frac{π}{2}$ by $1$ .

Phase Shift: $\frac{π}{2}$

Step 6

List the properties of the trigonometric function.

Amplitude: $\frac{2}{3}$

Period: $2 π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $2$

Step 7