Trigonometry Examples

$y = 4 \cos (- 2 x - \frac{π}{3})$

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.

$a = 4$

$b = - 2$

$c = \frac{π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

Find the period of $4 \cos (- 2 x - \frac{π}{3})$ .

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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 $- 2$ in the formula for period.

$\frac{2 π}{| - 2 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$\frac{2 π}{2}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.4.2

Divide $π$ by $1$ .

$π$

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: $\frac{\frac{π}{3}}{- 2}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 4.5

Multiply $\frac{π}{3} (- \frac{1}{2})$ .

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Step 4.5.1

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

Phase Shift: $- \frac{π}{3 \cdot 2}$

Step 4.5.2

Multiply $3$ by $2$ .

Phase Shift: $- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: None

Step 6