Trigonometry Examples

$y = - \sin (\frac{6 x}{5} - \frac{2 π}{3})$

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.

$a = - 1$

$b = \frac{6}{5}$

$c = \frac{2 π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $- \sin (\frac{6 x}{5} - \frac{2 π}{3})$ .

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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 $\frac{6}{5}$ in the formula for period.

$\frac{2 π}{| \frac{6}{5} |}$

Step 3.3

$\frac{6}{5}$ is approximately $1.2$ which is positive so remove the absolute value

$\frac{2 π}{\frac{6}{5}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{6}$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{6}$

Step 3.5.2

Factor $2$ out of $6$ .

$2 (π) \frac{5}{2 (3)}$

Step 3.5.3

Cancel the common factor.

$2 π \frac{5}{2 \cdot 3}$

Step 3.5.4

Rewrite the expression.

$π \frac{5}{3}$

Step 3.6

Combine $π$ and $\frac{5}{3}$ .

$\frac{π \cdot 5}{3}$

Step 3.7

Move $5$ to the left of $π$ .

$\frac{5 π}{3}$

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: $\frac{\frac{2 π}{3}}{\frac{6}{5}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{2 π}{3} \cdot \frac{5}{6}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Factor $2$ out of $2 π$ .

Phase Shift: $\frac{2 (π)}{3} \cdot \frac{5}{6}$

Step 4.4.2

Factor $2$ out of $6$ .

Phase Shift: $\frac{2 (π)}{3} \cdot \frac{5}{2 (3)}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{2 π}{3} \cdot \frac{5}{2 \cdot 3}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{π}{3} \cdot \frac{5}{3}$

Step 4.5

Multiply $\frac{π}{3}$ by $\frac{5}{3}$ .

Phase Shift: $\frac{π \cdot 5}{3 \cdot 3}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Multiply $3$ by $3$ .

Phase Shift: $\frac{π \cdot 5}{9}$

Step 4.6.2

Move $5$ to the left of $π$ .

Phase Shift: $\frac{5 π}{9}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{5 π}{3}$

Phase Shift: $\frac{5 π}{9}$ ( $\frac{5 π}{9}$ to the right)

Vertical Shift: None

Step 6