Trigonometry Examples

$y = 10 \cos (- \frac{θ}{2} + \frac{2 π}{3}) + 5$

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.

$a = 10$

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

$c = - \frac{2 π}{3}$

$d = 5$

Step 2

Find the amplitude $| a |$ .

Amplitude: $10$

Step 3

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

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Step 3.1

Find the period of $10 \cos (- \frac{θ}{2} + \frac{2 π}{3})$ .

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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 negative so negate $- \frac{1}{2}$ and 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 negative so negate $- \frac{1}{2}$ and 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: $\frac{- \frac{2 π}{3}}{- \frac{1}{2}}$

Step 4.3

Dividing two negative values results in a positive value.

Phase Shift: $\frac{\frac{2 π}{3}}{\frac{1}{2}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{2 π}{3} \cdot 2$

Step 4.5

Multiply $\frac{2 π}{3} \cdot 2$ .

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Step 4.5.1

Combine $\frac{2 π}{3}$ and $2$ .

Phase Shift: $\frac{2 π \cdot 2}{3}$

Step 4.5.2

Multiply $2$ by $2$ .

Phase Shift: $\frac{4 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $10$

Period: $4 π$

Phase Shift: $\frac{4 π}{3}$ ( $\frac{4 π}{3}$ to the right)

Vertical Shift: $5$

Step 6