Trigonometry Examples

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

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.

$a = 4$

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

$c = - \frac{π}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

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

$\frac{2 π}{| \frac{3}{4} |}$

Step 3.3

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{4}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{3}$

Step 3.5

Multiply $2 π \frac{4}{3}$ .

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Step 3.5.1

Combine $\frac{4}{3}$ and $2$ .

$\frac{4 \cdot 2}{3} π$

Step 3.5.2

Multiply $4$ by $2$ .

$\frac{8}{3} π$

Step 3.5.3

Combine $\frac{8}{3}$ and $π$ .

$\frac{8 π}{3}$

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: $\frac{- \frac{π}{4}}{\frac{3}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Cancel the common factor of $4$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

Phase Shift: $- π \frac{1}{3}$

Step 4.5

Combine $\frac{1}{3}$ and $π$ .

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{8 π}{3}$

Phase Shift: $- \frac{π}{3}$ ( $\frac{π}{3}$ to the left)

Vertical Shift: None

Step 6