Trigonometry Examples

$y = 2 \sin (\frac{8 π x}{3} + \frac{3 π}{2})$

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.

$a = 2$

$b = \frac{8 π}{3}$

$c = - \frac{3 π}{2}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

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

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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 $\frac{8 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{8 π}{3} |}$

Step 3.3

$\frac{8 π}{3}$ is approximately $8.3775804$ which is positive so remove the absolute value

$\frac{2 π}{\frac{8 π}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{8 π}$

Step 3.5

Cancel the common factor of $2 π$ .

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Step 3.5.1

Factor $2 π$ out of $8 π$ .

$2 π \frac{3}{2 π (4)}$

Step 3.5.2

Cancel the common factor.

$2 π \frac{3}{2 π \cdot 4}$

Step 3.5.3

Rewrite the expression.

$\frac{3}{4}$

Step 4

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

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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: $\frac{- \frac{3 π}{2}}{\frac{8 π}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{3 π}{2} \cdot \frac{3}{8 π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Move the leading negative in $- \frac{3 π}{2}$ into the numerator.

Phase Shift: $\frac{- 3 π}{2} \cdot \frac{3}{8 π}$

Step 4.4.2

Factor $π$ out of $- 3 π$ .

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{3}{8 π}$

Step 4.4.3

Factor $π$ out of $8 π$ .

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{3}{π \cdot 8}$

Step 4.4.4

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{3}{π \cdot 8}$

Step 4.4.5

Rewrite the expression.

Phase Shift: $\frac{- 3}{2} \cdot \frac{3}{8}$

Step 4.5

Multiply $\frac{- 3}{2}$ by $\frac{3}{8}$ .

Phase Shift: $\frac{- 3 \cdot 3}{2 \cdot 8}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Multiply $- 3$ by $3$ .

Phase Shift: $\frac{- 9}{2 \cdot 8}$

Step 4.6.2

Multiply $2$ by $8$ .

Phase Shift: $\frac{- 9}{16}$

Step 4.6.3

Move the negative in front of the fraction.

Phase Shift: $- \frac{9}{16}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{3}{4}$

Phase Shift: $- \frac{9}{16}$ ( $\frac{9}{16}$ to the left)

Vertical Shift: None

Step 6