Trigonometry Examples

$- 5 \sin (\frac{2 π}{5} x - 2 π) + 17$

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.

$a = - 5$

$b = \frac{2 π}{5}$

$c = 2 π$

$d = 17$

Step 2

Find the amplitude $| a |$ .

Amplitude: $5$

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

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

$\frac{2 π}{| \frac{2 π}{5} |}$

Step 3.1.3

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

$\frac{2 π}{\frac{2 π}{5}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2 π}$

Step 3.1.5

Cancel the common factor of $2 π$ .

Tap for more steps...

Step 3.1.5.1

Cancel the common factor.

$2 π \frac{5}{2 π}$

Step 3.1.5.2

Rewrite the expression.

$5$

Step 3.2

Find the period of $17$ .

Tap for more steps...

Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

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

$\frac{2 π}{| \frac{2 π}{5} |}$

Step 3.2.3

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

$\frac{2 π}{\frac{2 π}{5}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2 π}$

Step 3.2.5

Cancel the common factor of $2 π$ .

Tap for more steps...

Step 3.2.5.1

Cancel the common factor.

$2 π \frac{5}{2 π}$

Step 3.2.5.2

Rewrite the expression.

$5$

Step 3.3

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

$5$

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: $\frac{2 π}{\frac{2 π}{5}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $2 π (\frac{5}{2 π})$

Step 4.4

Cancel the common factor of $2 π$ .

Tap for more steps...

Step 4.4.1

Cancel the common factor.

Phase Shift: $2 π (\frac{5}{2 π})$

Step 4.4.2

Rewrite the expression.

Phase Shift: $5$

Step 5

List the properties of the trigonometric function.

Amplitude: $5$

Period: $5$

Phase Shift: $5$ ( $5$ to the right)

Vertical Shift: $17$

Step 6