Trigonometry Examples

$y = 6 \sin (\frac{2 π}{5} x + 4)$

Step 1

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

$a = 6$

$b = \frac{2 π}{5}$

$c = - 4$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $6$

Step 3

Find the period of $6 \sin (\frac{2 π x}{5} + 4)$ .

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 $\frac{2 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{5} |}$

Step 3.3

$\frac{2 π}{5}$ is approximately $1.25663706$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{5}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2 π}$

Step 3.5

Cancel the common factor of $2 π$ .

Tap for more steps...

Step 3.5.1

Cancel the common factor.

$2 π \frac{5}{2 π}$

Step 3.5.2

Rewrite the expression.

$5$

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: $\frac{- 4}{\frac{2 π}{5}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- 4 \frac{5}{2 π}$

Step 4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.1

Factor $2$ out of $- 4$ .

Phase Shift: $2 (- 2) (\frac{5}{2 π})$

Step 4.4.2

Factor $2$ out of $2 π$ .

Phase Shift: $2 (- 2) (\frac{5}{2 (π)})$

Step 4.4.3

Cancel the common factor.

Phase Shift: $2 \cdot (- 2 \frac{5}{2 π})$

Step 4.4.4

Rewrite the expression.

Phase Shift: $- 2 \frac{5}{π}$

Step 4.5

Combine $- 2$ and $\frac{5}{π}$ .

Phase Shift: $\frac{- 2 \cdot 5}{π}$

Step 4.6

Simplify the expression.

Tap for more steps...

Step 4.6.1

Multiply $- 2$ by $5$ .

Phase Shift: $\frac{- 10}{π}$

Step 4.6.2

Move the negative in front of the fraction.

Phase Shift: $- \frac{10}{π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $6$

Period: $5$

Phase Shift: $- \frac{10}{π}$ ( $\frac{10}{π}$ to the left)

Vertical Shift: None

Step 6