Trigonometry Examples

$f (x) = 4 \sin (\frac{1}{π} x - 2) + 8$

Step 1

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

$a = 4$

$b = \frac{1}{π}$

$c = 2$

$d = 8$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

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Step 3.1

Find the period of $4 \sin (\frac{x}{π} - 2)$ .

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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 $\frac{1}{π}$ in the formula for period.

$\frac{2 π}{| \frac{1}{π} |}$

Step 3.1.3

$\frac{1}{π}$ is approximately $0.31830988$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{π}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π π$

Step 3.1.5

Multiply $2 π π$ .

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Step 3.1.5.1

Raise $π$ to the power of $1$ .

$2 (π^{1} π)$

Step 3.1.5.2

Raise $π$ to the power of $1$ .

$2 (π^{1} π^{1})$

Step 3.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 π^{1 + 1}$

Step 3.1.5.4

Add $1$ and $1$ .

$2 π^{2}$

Step 3.2

Find the period of $8$ .

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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 $\frac{1}{π}$ in the formula for period.

$\frac{2 π}{| \frac{1}{π} |}$

Step 3.2.3

$\frac{1}{π}$ is approximately $0.31830988$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{π}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π π$

Step 3.2.5

Multiply $2 π π$ .

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Step 3.2.5.1

Raise $π$ to the power of $1$ .

$2 (π^{1} π)$

Step 3.2.5.2

Raise $π$ to the power of $1$ .

$2 (π^{1} π^{1})$

Step 3.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 π^{1 + 1}$

Step 3.2.5.4

Add $1$ and $1$ .

$2 π^{2}$

Step 3.3

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

$2 π^{2}$

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: $\frac{2}{\frac{1}{π}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $2 π$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $2 π^{2}$

Phase Shift: $2 π$ ( $2 π$ to the right)

Vertical Shift: $8$

Step 6