Trigonometry Examples

$y = 4 \cos (\frac{4 π x}{7} + 5)$

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.

$a = 4$

$b = \frac{4 π}{7}$

$c = - 5$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

Find the period of $4 \cos (\frac{4 π x}{7} + 5)$ .

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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 $\frac{4 π}{7}$ in the formula for period.

$\frac{2 π}{| \frac{4 π}{7} |}$

Step 3.3

$\frac{4 π}{7}$ is approximately $1.7951958$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4 π}{7}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{4 π}$

Step 3.5

Cancel the common factor of $2 π$ .

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Step 3.5.1

Factor $2 π$ out of $4 π$ .

$2 π \frac{7}{2 π (2)}$

Step 3.5.2

Cancel the common factor.

$2 π \frac{7}{2 π \cdot 2}$

Step 3.5.3

Rewrite the expression.

$\frac{7}{2}$

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: $\frac{- 5}{\frac{4 π}{7}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- 5 \frac{7}{4 π}$

Step 4.4

Multiply $- 5 \frac{7}{4 π}$ .

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Step 4.4.1

Combine $- 5$ and $\frac{7}{4 π}$ .

Phase Shift: $\frac{- 5 \cdot 7}{4 π}$

Step 4.4.2

Multiply $- 5$ by $7$ .

Phase Shift: $\frac{- 35}{4 π}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{35}{4 π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{7}{2}$

Phase Shift: $- \frac{35}{4 π}$ ( $\frac{35}{4 π}$ to the left)

Vertical Shift: None

Step 6