Trigonometry Examples

$y = - \frac{1}{2} \cdot \cos (π x - \frac{π}{18})$

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.

$a = - \frac{1}{2}$

$b = π$

$c = \frac{π}{18}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

Step 3

Find the period of $- \frac{\cos (π x - \frac{π}{18})}{2}$ .

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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: $\frac{\frac{π}{18}}{π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{18} \cdot \frac{1}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Cancel the common factor.

Phase Shift: $\frac{π}{18} \cdot \frac{1}{π}$

Step 4.4.2

Rewrite the expression.

Phase Shift: $\frac{1}{18}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $2$

Phase Shift: $\frac{1}{18}$ ( $\frac{1}{18}$ to the right)

Vertical Shift: None

Step 6