Trigonometry Examples

$y = 2 \cos (\frac{1}{2} x) + 1$

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.

$a = 2$

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

$c = 0$

$d = 1$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$2$

Step 3

Find the period using the formula

$\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of

$2 \cos (\frac{x}{2})$ .

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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

$1$ .

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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:

$\frac{0}{\frac{1}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$0 \cdot 2$

Step 4.4

Multiply

$0$ by

$2$ .

Phase Shift:

$0$

Phase Shift:

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

$2$

Period:

$4 π$

Phase Shift: None

Vertical Shift:

$1$

Step 6