Trigonometry Examples

$y = \sin (\frac{4 x}{3} + \frac{4}{3})$

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.

$a = 1$

$b = \frac{4}{3}$

$c = - \frac{4}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $\sin (\frac{4 x}{3} + \frac{4}{3})$ .

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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 $\frac{4}{3}$ in the formula for period.

$\frac{2 π}{| \frac{4}{3} |}$

Step 3.3

$\frac{4}{3}$ is approximately $1.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{4}$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{4}$

Step 3.5.2

Factor $2$ out of $4$ .

$2 (π) \frac{3}{2 (2)}$

Step 3.5.3

Cancel the common factor.

$2 π \frac{3}{2 \cdot 2}$

Step 3.5.4

Rewrite the expression.

$π \frac{3}{2}$

Step 3.6

Combine $π$ and $\frac{3}{2}$ .

$\frac{π \cdot 3}{2}$

Step 3.7

Move $3$ to the left of $π$ .

$\frac{3 π}{2}$

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

Step 4.3

Cancel the common factor of $\frac{4}{3}$ .

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Step 4.3.1

Cancel the common factor.

Phase Shift: $\frac{- \frac{4}{3}}{\frac{4}{3}}$

Step 4.3.2

Divide $- 1$ by $1$ .

Phase Shift: $- 1$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{3 π}{2}$

Phase Shift: $- 1$ ( $1$ to the left)

Vertical Shift: None

Step 6