Trigonometry Examples

$f (x) = 4 \cos (7 x + \frac{7 π}{4})$

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.

$a = 4$

$b = 7$

$c = - \frac{7 π}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

Find the period of $4 \cos (7 x + \frac{7 π}{4})$ .

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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 $7$ in the formula for period.

$\frac{2 π}{| 7 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$\frac{2 π}{7}$

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: $\frac{- \frac{7 π}{4}}{7}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{7 π}{4} \cdot \frac{1}{7}$

Step 4.4

Cancel the common factor of $7$ .

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Step 4.4.1

Move the leading negative in $- \frac{7 π}{4}$ into the numerator.

Phase Shift: $\frac{- 7 π}{4} \cdot \frac{1}{7}$

Step 4.4.2

Factor $7$ out of $- 7 π$ .

Phase Shift: $\frac{7 (- π)}{4} \cdot \frac{1}{7}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{7 (- π)}{4} \cdot \frac{1}{7}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- π}{4}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{2 π}{7}$

Phase Shift: $- \frac{π}{4}$ ( $\frac{π}{4}$ to the left)

Vertical Shift: None

Step 6