Trigonometry Examples

$y = 6 \cos (\frac{θ}{2} + \frac{π}{6}) - 5$

Step 1

Use the form $a \cos (b θ - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 6$

$b = \frac{1}{2}$

$c = - \frac{π}{6}$

$d = - 5$

Step 2

Find the amplitude $| a |$ .

Amplitude: $6$

Step 3

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of $6 \cos (\frac{θ}{2} + \frac{π}{6})$ .

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Step 3.1.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.1.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.1.5

Multiply $2$ by $2$ .

$4 π$

Step 3.2

Find the period of $- 5$ .

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Step 3.2.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.2.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.2.5

Multiply $2$ by $2$ .

$4 π$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$4 π$

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: $\frac{- \frac{π}{6}}{\frac{1}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot 2$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

Phase Shift: $\frac{- π}{6} \cdot 2$

Step 4.4.2

Factor $2$ out of $6$ .

Phase Shift: $\frac{- π}{2 (3)} \cdot 2$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{- π}{2 \cdot 3} \cdot 2$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- π}{3}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $6$

Period: $4 π$

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

Vertical Shift: $- 5$

Step 6