Trigonometry Examples

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

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.

$a = \frac{1}{3}$

$b = 1$

$c = - \frac{π}{6}$

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

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{3}$

Step 3

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

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Step 3.1

Find the period of $\frac{\sin (x + \frac{π}{6})}{3}$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.1.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2

Find the period of $- \frac{1}{2}$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.2.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.3

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

$2 π$

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: $\frac{- \frac{π}{6}}{1}$

Step 4.3

Divide $- \frac{π}{6}$ by $1$ .

Phase Shift: $- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{3}$

Period: $2 π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: $- \frac{1}{2}$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - \frac{π}{6}$ .

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Step 6.1.1

Replace the variable $x$ with $- \frac{π}{6}$ in the expression.

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Add $- \frac{π}{6}$ and $\frac{π}{6}$ .

$f (- \frac{π}{6}) = \frac{\sin (0)}{3} - \frac{1}{2}$

Step 6.1.2.2

Simplify each term.

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Step 6.1.2.2.1

The exact value of $\sin (0)$ is $0$ .

$f (- \frac{π}{6}) = \frac{0}{3} - \frac{1}{2}$

Step 6.1.2.2.2

Divide $0$ by $3$ .

$f (- \frac{π}{6}) = 0 - \frac{1}{2}$

Step 6.1.2.3

Subtract $\frac{1}{2}$ from $0$ .

$f (- \frac{π}{6}) = - \frac{1}{2}$

Step 6.1.2.4

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 6.2

Find the point at $x = \frac{π}{3}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{π}{3}$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{π}{3} \cdot \frac{2}{2} + \frac{π}{6})}{3} - \frac{1}{2}$

Step 6.2.2.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.2.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{π \cdot 2}{3 \cdot 2} + \frac{π}{6})}{3} - \frac{1}{2}$

Step 6.2.2.2.2

Multiply $3$ by $2$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{π \cdot 2}{6} + \frac{π}{6})}{3} - \frac{1}{2}$

Step 6.2.2.3

Combine the numerators over the common denominator.

$f (\frac{π}{3}) = \frac{\sin (\frac{π \cdot 2 + π}{6})}{3} - \frac{1}{2}$

Step 6.2.2.4

Add $π \cdot 2$ and $π$ .

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Step 6.2.2.4.1

Reorder $π$ and $2$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{2 \cdot π + π}{6})}{3} - \frac{1}{2}$

Step 6.2.2.4.2

Add $2 \cdot π$ and $π$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{3 \cdot π}{6})}{3} - \frac{1}{2}$

Step 6.2.2.5

Simplify each term.

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Step 6.2.2.5.1

Cancel the common factor of $3$ and $6$ .

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Step 6.2.2.5.1.1

Factor $3$ out of $3 \cdot π$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{3 (π)}{6})}{3} - \frac{1}{2}$

Step 6.2.2.5.1.2

Cancel the common factors.

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Step 6.2.2.5.1.2.1

Factor $3$ out of $6$ .

$f (\frac{π}{3}) = \frac{\sin (\frac{3 π}{3 \cdot 2})}{3} - \frac{1}{2}$

Step 6.2.2.5.1.2.2

Cancel the common factor.

$f (\frac{π}{3}) = \frac{\sin (\frac{3 π}{3 \cdot 2})}{3} - \frac{1}{2}$

Step 6.2.2.5.1.2.3

Rewrite the expression.

$f (\frac{π}{3}) = \frac{\sin (\frac{π}{2})}{3} - \frac{1}{2}$

Step 6.2.2.5.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{π}{3}) = \frac{1}{3} - \frac{1}{2}$

Step 6.2.2.6

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{π}{3}) = \frac{1}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 6.2.2.7

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{π}{3}) = \frac{1}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.2.2.8

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.8.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$f (\frac{π}{3}) = \frac{2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.2.2.8.2

Multiply $3$ by $2$ .

$f (\frac{π}{3}) = \frac{2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.2.2.8.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$f (\frac{π}{3}) = \frac{2}{6} - \frac{3}{2 \cdot 3}$

Step 6.2.2.8.4

Multiply $2$ by $3$ .

$f (\frac{π}{3}) = \frac{2}{6} - \frac{3}{6}$

Step 6.2.2.9

Combine the numerators over the common denominator.

$f (\frac{π}{3}) = \frac{2 - 3}{6}$

Step 6.2.2.10

Subtract $3$ from $2$ .

$f (\frac{π}{3}) = \frac{- 1}{6}$

Step 6.2.2.11

Move the negative in front of the fraction.

$f (\frac{π}{3}) = - \frac{1}{6}$

Step 6.2.2.12

The final answer is $- \frac{1}{6}$ .

$- \frac{1}{6}$

Step 6.3

Find the point at $x = \frac{5 π}{6}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{5 π}{6}$ in the expression.

$f (\frac{5 π}{6}) = \frac{\sin ((\frac{5 π}{6}) + \frac{π}{6})}{3} - \frac{1}{2}$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify by adding terms.

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Step 6.3.2.1.1

Combine the numerators over the common denominator.

$f (\frac{5 π}{6}) = \frac{\sin (\frac{5 π + π}{6})}{3} - \frac{1}{2}$

Step 6.3.2.1.2

Add $5 π$ and $π$ .

$f (\frac{5 π}{6}) = \frac{\sin (\frac{6 π}{6})}{3} - \frac{1}{2}$

Step 6.3.2.2

Simplify each term.

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Step 6.3.2.2.1

Cancel the common factor of $6$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$f (\frac{5 π}{6}) = \frac{\sin (\frac{6 π}{6})}{3} - \frac{1}{2}$

Step 6.3.2.2.1.2

Divide $π$ by $1$ .

$f (\frac{5 π}{6}) = \frac{\sin (π)}{3} - \frac{1}{2}$

Step 6.3.2.2.2

Simplify the numerator.

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Step 6.3.2.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (\frac{5 π}{6}) = \frac{\sin (0)}{3} - \frac{1}{2}$

Step 6.3.2.2.2.2

The exact value of $\sin (0)$ is $0$ .

$f (\frac{5 π}{6}) = \frac{0}{3} - \frac{1}{2}$

Step 6.3.2.2.3

Divide $0$ by $3$ .

$f (\frac{5 π}{6}) = 0 - \frac{1}{2}$

Step 6.3.2.3

Subtract $\frac{1}{2}$ from $0$ .

$f (\frac{5 π}{6}) = - \frac{1}{2}$

Step 6.3.2.4

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 6.4

Find the point at $x = \frac{4 π}{3}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{4 π}{3}$ in the expression.

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.4.2.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.2.1

Multiply $\frac{4 π}{3}$ by $\frac{2}{2}$ .

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

Step 6.4.2.2.2

Multiply $3$ by $2$ .

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

Step 6.4.2.3

Combine the numerators over the common denominator.

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

Step 6.4.2.4

Simplify each term.

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Step 6.4.2.4.1

Simplify the numerator.

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Step 6.4.2.4.1.1

Multiply $2$ by $4$ .

$f (\frac{4 π}{3}) = \frac{\sin (\frac{8 π + π}{6})}{3} - \frac{1}{2}$

Step 6.4.2.4.1.2

Add $8 π$ and $π$ .

$f (\frac{4 π}{3}) = \frac{\sin (\frac{9 π}{6})}{3} - \frac{1}{2}$

Step 6.4.2.4.2

Simplify the numerator.

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Step 6.4.2.4.2.1

Cancel the common factor of $9$ and $6$ .

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Step 6.4.2.4.2.1.1

Factor $3$ out of $9 π$ .

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

Step 6.4.2.4.2.1.2

Cancel the common factors.

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Step 6.4.2.4.2.1.2.1

Factor $3$ out of $6$ .

$f (\frac{4 π}{3}) = \frac{\sin (\frac{3 (3 π)}{3 (2)})}{3} - \frac{1}{2}$

Step 6.4.2.4.2.1.2.2

Cancel the common factor.

$f (\frac{4 π}{3}) = \frac{\sin (\frac{3 (3 π)}{3 \cdot 2})}{3} - \frac{1}{2}$

Step 6.4.2.4.2.1.2.3

Rewrite the expression.

$f (\frac{4 π}{3}) = \frac{\sin (\frac{3 π}{2})}{3} - \frac{1}{2}$

Step 6.4.2.4.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{4 π}{3}) = \frac{- \sin (\frac{π}{2})}{3} - \frac{1}{2}$

Step 6.4.2.4.2.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{4 π}{3}) = \frac{- 1 \cdot 1}{3} - \frac{1}{2}$

Step 6.4.2.4.2.4

Multiply $- 1$ by $1$ .

$f (\frac{4 π}{3}) = \frac{- 1}{3} - \frac{1}{2}$

Step 6.4.2.4.3

Move the negative in front of the fraction.

$f (\frac{4 π}{3}) = - \frac{1}{3} - \frac{1}{2}$

Step 6.4.2.5

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{4 π}{3}) = - \frac{1}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 6.4.2.6

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{4 π}{3}) = - \frac{1}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.4.2.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.7.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$f (\frac{4 π}{3}) = - \frac{2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.4.2.7.2

Multiply $3$ by $2$ .

$f (\frac{4 π}{3}) = - \frac{2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.4.2.7.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$f (\frac{4 π}{3}) = - \frac{2}{6} - \frac{3}{2 \cdot 3}$

Step 6.4.2.7.4

Multiply $2$ by $3$ .

$f (\frac{4 π}{3}) = - \frac{2}{6} - \frac{3}{6}$

Step 6.4.2.8

Combine the numerators over the common denominator.

$f (\frac{4 π}{3}) = \frac{- 2 - 3}{6}$

Step 6.4.2.9

Subtract $3$ from $- 2$ .

$f (\frac{4 π}{3}) = \frac{- 5}{6}$

Step 6.4.2.10

Move the negative in front of the fraction.

$f (\frac{4 π}{3}) = - \frac{5}{6}$

Step 6.4.2.11

The final answer is $- \frac{5}{6}$ .

$- \frac{5}{6}$

Step 6.5

Find the point at $x = \frac{11 π}{6}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{11 π}{6}$ in the expression.

$f (\frac{11 π}{6}) = \frac{\sin ((\frac{11 π}{6}) + \frac{π}{6})}{3} - \frac{1}{2}$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify by adding terms.

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Step 6.5.2.1.1

Combine the numerators over the common denominator.

$f (\frac{11 π}{6}) = \frac{\sin (\frac{11 π + π}{6})}{3} - \frac{1}{2}$

Step 6.5.2.1.2

Add $11 π$ and $π$ .

$f (\frac{11 π}{6}) = \frac{\sin (\frac{12 π}{6})}{3} - \frac{1}{2}$

Step 6.5.2.2

Simplify each term.

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Step 6.5.2.2.1

Cancel the common factor of $12$ and $6$ .

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Step 6.5.2.2.1.1

Factor $6$ out of $12 π$ .

$f (\frac{11 π}{6}) = \frac{\sin (\frac{6 (2 π)}{6})}{3} - \frac{1}{2}$

Step 6.5.2.2.1.2

Cancel the common factors.

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Step 6.5.2.2.1.2.1

Factor $6$ out of $6$ .

$f (\frac{11 π}{6}) = \frac{\sin (\frac{6 (2 π)}{6 (1)})}{3} - \frac{1}{2}$

Step 6.5.2.2.1.2.2

Cancel the common factor.

$f (\frac{11 π}{6}) = \frac{\sin (\frac{6 (2 π)}{6 \cdot 1})}{3} - \frac{1}{2}$

Step 6.5.2.2.1.2.3

Rewrite the expression.

$f (\frac{11 π}{6}) = \frac{\sin (\frac{2 π}{1})}{3} - \frac{1}{2}$

Step 6.5.2.2.1.2.4

Divide $2 π$ by $1$ .

$f (\frac{11 π}{6}) = \frac{\sin (2 π)}{3} - \frac{1}{2}$

Step 6.5.2.2.2

Simplify the numerator.

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Step 6.5.2.2.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{11 π}{6}) = \frac{\sin (0)}{3} - \frac{1}{2}$

Step 6.5.2.2.2.2

The exact value of $\sin (0)$ is $0$ .

$f (\frac{11 π}{6}) = \frac{0}{3} - \frac{1}{2}$

Step 6.5.2.2.3

Divide $0$ by $3$ .

$f (\frac{11 π}{6}) = 0 - \frac{1}{2}$

Step 6.5.2.3

Subtract $\frac{1}{2}$ from $0$ .

$f (\frac{11 π}{6}) = - \frac{1}{2}$

Step 6.5.2.4

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{6} & - \frac{1}{2} \\ \frac{π}{3} & - \frac{1}{6} \\ \frac{5 π}{6} & - \frac{1}{2} \\ \frac{4 π}{3} & - \frac{5}{6} \\ \frac{11 π}{6} & - \frac{1}{2} \end{array}$

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: $\frac{1}{3}$

Period: $2 π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: $- \frac{1}{2}$

Step 8