Trigonometry Examples

$g (x) = \cos (\frac{2 π}{3} x) + 1$

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.

$a = 1$

$b = \frac{2 π}{3}$

$c = 0$

$d = 1$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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Step 3.1

Find the period of $\cos (\frac{2 π x}{3})$ .

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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 $\frac{2 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{3} |}$

Step 3.1.3

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Step 3.1.5

Cancel the common factor of $2 π$ .

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Step 3.1.5.1

Cancel the common factor.

$2 π \frac{3}{2 π}$

Step 3.1.5.2

Rewrite the expression.

$3$

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 $\frac{2 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{3} |}$

Step 3.2.3

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Step 3.2.5

Cancel the common factor of $2 π$ .

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Step 3.2.5.1

Cancel the common factor.

$2 π \frac{3}{2 π}$

Step 3.2.5.2

Rewrite the expression.

$3$

Step 3.3

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

$3$

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: $\frac{0}{\frac{2 π}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{2 π})$

Step 4.4

Multiply $0$ by $\frac{3}{2 π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $3$

Phase Shift: None

Vertical Shift: $1$

Step 6