Trigonometry Examples

$y = \cos (3 π x - \frac{2 π}{3})$

Step 1

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

$a = 1$

$b = 3 π$

$c = \frac{2 π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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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 $3 π$ in the formula for period.

$\frac{2 π}{| 3 π |}$

Step 3.3

$3 π$ is approximately $9.42477796$ which is positive so remove the absolute value

$\frac{2 π}{3 π}$

Step 3.4

Cancel the common factor of $π$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{3 π}$

Step 3.4.2

Rewrite the expression.

$\frac{2}{3}$

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: $\frac{\frac{2 π}{3}}{3 π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{2 π}{3} \cdot \frac{1}{3 π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Factor $π$ out of $2 π$ .

Phase Shift: $\frac{π \cdot 2}{3} \cdot \frac{1}{3 π}$

Step 4.4.2

Factor $π$ out of $3 π$ .

Phase Shift: $\frac{π \cdot 2}{3} \cdot \frac{1}{π \cdot 3}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{π \cdot 2}{3} \cdot \frac{1}{π \cdot 3}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{2}{3} \cdot \frac{1}{3}$

Step 4.5

Multiply $\frac{2}{3}$ by $\frac{1}{3}$ .

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

Step 4.6

Multiply $3$ by $3$ .

Phase Shift: $\frac{2}{9}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2}{3}$

Phase Shift: $\frac{2}{9}$ ( $\frac{2}{9}$ to the right)

Vertical Shift: None

Step 6