Trigonometry Examples

$y = \frac{1}{2} \sin (6 θ - \frac{3 π}{4}) + 3$

Step 1

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

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

$b = 6$

$c = \frac{3 π}{4}$

$d = 3$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

Step 3

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

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Step 3.1

Find the period of $\frac{\sin (6 θ - \frac{3 π}{4})}{2}$ .

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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 $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 3.1.4

Cancel the common factor of $2$ and $6$ .

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Step 3.1.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 3.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 3.1.4.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 3.2

Find the period of $3$ .

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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 $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 3.2.4

Cancel the common factor of $2$ and $6$ .

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Step 3.2.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 3.3

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

$\frac{π}{3}$

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: $\frac{\frac{3 π}{4}}{6}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π}{4} \cdot \frac{1}{6}$

Step 4.4

Cancel the common factor of $3$ .

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Step 4.4.1

Factor $3$ out of $3 π$ .

Phase Shift: $\frac{3 (π)}{4} \cdot \frac{1}{6}$

Step 4.4.2

Factor $3$ out of $6$ .

Phase Shift: $\frac{3 (π)}{4} \cdot \frac{1}{3 (2)}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{3 π}{4} \cdot \frac{1}{3 \cdot 2}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{π}{4} \cdot \frac{1}{2}$

Step 4.5

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

Phase Shift: $\frac{π}{4 \cdot 2}$

Step 4.6

Multiply $4$ by $2$ .

Phase Shift: $\frac{π}{8}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $\frac{π}{3}$

Phase Shift: $\frac{π}{8}$ ( $\frac{π}{8}$ to the right)

Vertical Shift: $3$

Step 6