Trigonometry Examples

$y = 2 \cos (\frac{8 x}{3} - 2 π)$

Step 1

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

$a = 2$

$b = \frac{8}{3}$

$c = 2 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period of $2 \cos (\frac{8 x}{3} - 2 π)$ .

Tap for more steps...

Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{8}{3}$ in the formula for period.

$\frac{2 π}{| \frac{8}{3} |}$

Step 3.3

$\frac{8}{3}$ is approximately $2.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{8}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{8}$

Step 3.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{8}$

Step 3.5.2

Factor $2$ out of $8$ .

$2 (π) \frac{3}{2 (4)}$

Step 3.5.3

Cancel the common factor.

$2 π \frac{3}{2 \cdot 4}$

Step 3.5.4

Rewrite the expression.

$π \frac{3}{4}$

Step 3.6

Combine $π$ and $\frac{3}{4}$ .

$\frac{π \cdot 3}{4}$

Step 3.7

Move $3$ to the left of $π$ .

$\frac{3 π}{4}$

Step 4

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

Tap for more steps...

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: $\frac{2 π}{\frac{8}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $2 π (\frac{3}{8})$

Step 4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.1

Factor $2$ out of $2 π$ .

Phase Shift: $2 (π) (\frac{3}{8})$

Step 4.4.2

Factor $2$ out of $8$ .

Phase Shift: $2 (π) (\frac{3}{2 (4)})$

Step 4.4.3

Cancel the common factor.

Phase Shift: $2 π (\frac{3}{2 \cdot 4})$

Step 4.4.4

Rewrite the expression.

Phase Shift: $π (\frac{3}{4})$

Step 4.5

Combine $π$ and $\frac{3}{4}$ .

Phase Shift: $\frac{π \cdot 3}{4}$

Step 4.6

Move $3$ to the left of $π$ .

Phase Shift: $\frac{3 π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{3 π}{4}$

Phase Shift: $\frac{3 π}{4}$ ( $\frac{3 π}{4}$ to the right)

Vertical Shift: None

Step 6