Trigonometry Examples

$y = 4 \cos (π (x + \frac{1}{3}))$

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.

$a = 4$

$b = π$

$c = - \frac{π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

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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 $π$ in the formula for period.

$\frac{2 π}{| π |}$

Step 3.3

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Step 3.4

Cancel the common factor of $π$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 3.4.2

Divide $2$ by $1$ .

$2$

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: $\frac{- \frac{π}{3}}{π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{3}$ into the numerator.

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

Step 4.4.2

Factor $π$ out of $- π$ .

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

Step 4.4.3

Cancel the common factor.

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

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- 1}{3}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $2$

Phase Shift: $- \frac{1}{3}$ ( $\frac{1}{3}$ to the left)

Vertical Shift: None

Step 6