Trigonometry Examples

$y = - 2 \cos (8 (x + \frac{π}{2}))$

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.

$a = - 2$

$b = 8$

$c = - 4 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period of $- 2 \cos (8 x + 4 π)$ .

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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 $8$ in the formula for period.

$\frac{2 π}{| 8 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{2 π}{8}$

Step 3.4

Cancel the common factor of $2$ and $8$ .

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Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{8}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $8$ .

$\frac{2 π}{2 \cdot 4}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 4}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{4}$

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: $\frac{- 4 π}{8}$

Step 4.3

Cancel the common factor of $- 4$ and $8$ .

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Step 4.3.1

Factor $4$ out of $- 4 π$ .

Phase Shift: $\frac{4 (- π)}{8}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $4$ out of $8$ .

Phase Shift: $\frac{4 (- π)}{4 (2)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{4 (- π)}{4 \cdot 2}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{- π}{2}$

Step 4.4

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{π}{4}$

Phase Shift: $- \frac{π}{2}$ ( $\frac{π}{2}$ to the left)

Vertical Shift: None

Step 6