Trigonometry Examples

$y = 0.5 \sin (2 π x) + \frac{2}{3}$

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.

$a = 0.5$

$b = 2 π$

$c = 0$

$d = \frac{2}{3}$

Step 2

Find the amplitude $| a |$ .

Amplitude: $0.5$

Step 3

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

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Step 3.1

Find the period of $0.5 \sin (2 π x)$ .

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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 $2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 3.1.3

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 3.1.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 3.1.5.2

Rewrite the expression.

$1$

Step 3.2

Find the period of $\frac{2}{3}$ .

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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 $2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 3.2.3

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 3.2.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 3.2.5.2

Rewrite the expression.

$1$

Step 3.3

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

$1$

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: $\frac{0}{2 π}$

Step 4.3

Cancel the common factor of $0$ and $2$ .

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Step 4.3.1

Factor $2$ out of $0$ .

Phase Shift: $\frac{2 (0)}{2 π}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2 π$ .

Phase Shift: $\frac{2 (0)}{2 (π)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{2 \cdot 0}{2 π}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{0}{π}$

Step 4.4

Divide $0$ by $π$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $0.5$

Period: $1$

Phase Shift: None

Vertical Shift: $\frac{2}{3}$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 0.5 \sin (2 π (0)) + \frac{2}{3}$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Multiply $2 π (0)$ .

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Step 6.1.2.1.1.1

Multiply $0$ by $2$ .

$f (0) = 0.5 \sin (0 π) + \frac{2}{3}$

Step 6.1.2.1.1.2

Multiply $0$ by $π$ .

$f (0) = 0.5 \sin (0) + \frac{2}{3}$

Step 6.1.2.1.2

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

$f (0) = 0.5 \cdot 0 + \frac{2}{3}$

Step 6.1.2.1.3

Multiply $0.5$ by $0$ .

$f (0) = 0 + \frac{2}{3}$

Step 6.1.2.2

Add $0$ and $\frac{2}{3}$ .

$f (0) = \frac{2}{3}$

Step 6.1.2.3

The final answer is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 6.2

Find the point at $x = \frac{1}{4}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{1}{4}$ in the expression.

$f (\frac{1}{4}) = 0.5 \sin (2 π (\frac{1}{4})) + \frac{2}{3}$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.1

Factor $2$ out of $2 π$ .

$f (\frac{1}{4}) = 0.5 \sin (2 (π) (\frac{1}{4})) + \frac{2}{3}$

Step 6.2.2.1.1.2

Factor $2$ out of $4$ .

$f (\frac{1}{4}) = 0.5 \sin (2 (π) (\frac{1}{2 (2)})) + \frac{2}{3}$

Step 6.2.2.1.1.3

Cancel the common factor.

$f (\frac{1}{4}) = 0.5 \sin (2 π (\frac{1}{2 \cdot 2})) + \frac{2}{3}$

Step 6.2.2.1.1.4

Rewrite the expression.

$f (\frac{1}{4}) = 0.5 \sin (π (\frac{1}{2})) + \frac{2}{3}$

Step 6.2.2.1.2

Combine $π$ and $\frac{1}{2}$ .

$f (\frac{1}{4}) = 0.5 \sin (\frac{π}{2}) + \frac{2}{3}$

Step 6.2.2.1.3

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

$f (\frac{1}{4}) = 0.5 \cdot 1 + \frac{2}{3}$

Step 6.2.2.1.4

Multiply $0.5$ by $1$ .

$f (\frac{1}{4}) = 0.5 + \frac{2}{3}$

Step 6.2.2.2

To write $0.5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{1}{4}) = 0.5 \cdot \frac{3}{3} + \frac{2}{3}$

Step 6.2.2.3

Combine fractions.

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Step 6.2.2.3.1

Combine $0.5$ and $\frac{3}{3}$ .

$f (\frac{1}{4}) = \frac{0.5 \cdot 3}{3} + \frac{2}{3}$

Step 6.2.2.3.2

Combine the numerators over the common denominator.

$f (\frac{1}{4}) = \frac{0.5 \cdot 3 + 2}{3}$

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Multiply $0.5$ by $3$ .

$f (\frac{1}{4}) = \frac{1.5 + 2}{3}$

Step 6.2.2.4.2

Add $1.5$ and $2$ .

$f (\frac{1}{4}) = \frac{3.5}{3}$

Step 6.2.2.5

Divide $3.5$ by $3$ .

$f (\frac{1}{4}) = 1.1\overline{6}$

Step 6.2.2.6

The final answer is $1.1\overline{6}$ .

$1.1\overline{6}$

Step 6.3

Find the point at $x = \frac{1}{2}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = 0.5 \sin (2 π (\frac{1}{2})) + \frac{2}{3}$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.1

Factor $2$ out of $2 π$ .

$f (\frac{1}{2}) = 0.5 \sin (2 (π) (\frac{1}{2})) + \frac{2}{3}$

Step 6.3.2.1.1.2

Cancel the common factor.

$f (\frac{1}{2}) = 0.5 \sin (2 π (\frac{1}{2})) + \frac{2}{3}$

Step 6.3.2.1.1.3

Rewrite the expression.

$f (\frac{1}{2}) = 0.5 \sin (π) + \frac{2}{3}$

Step 6.3.2.1.2

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

$f (\frac{1}{2}) = 0.5 \sin (0) + \frac{2}{3}$

Step 6.3.2.1.3

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

$f (\frac{1}{2}) = 0.5 \cdot 0 + \frac{2}{3}$

Step 6.3.2.1.4

Multiply $0.5$ by $0$ .

$f (\frac{1}{2}) = 0 + \frac{2}{3}$

Step 6.3.2.2

Add $0$ and $\frac{2}{3}$ .

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

Step 6.3.2.3

The final answer is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 6.4

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

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Step 6.4.1

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

$f (\frac{3}{4}) = 0.5 \sin (2 π (\frac{3}{4})) + \frac{2}{3}$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Cancel the common factor of $2$ .

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Step 6.4.2.1.1.1

Factor $2$ out of $2 π$ .

$f (\frac{3}{4}) = 0.5 \sin (2 (π) (\frac{3}{4})) + \frac{2}{3}$

Step 6.4.2.1.1.2

Factor $2$ out of $4$ .

$f (\frac{3}{4}) = 0.5 \sin (2 (π) (\frac{3}{2 (2)})) + \frac{2}{3}$

Step 6.4.2.1.1.3

Cancel the common factor.

$f (\frac{3}{4}) = 0.5 \sin (2 π (\frac{3}{2 \cdot 2})) + \frac{2}{3}$

Step 6.4.2.1.1.4

Rewrite the expression.

$f (\frac{3}{4}) = 0.5 \sin (π (\frac{3}{2})) + \frac{2}{3}$

Step 6.4.2.1.2

Combine $π$ and $\frac{3}{2}$ .

$f (\frac{3}{4}) = 0.5 \sin (\frac{π \cdot 3}{2}) + \frac{2}{3}$

Step 6.4.2.1.3

Move $3$ to the left of $π$ .

$f (\frac{3}{4}) = 0.5 \sin (\frac{3 \cdot π}{2}) + \frac{2}{3}$

Step 6.4.2.1.4

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.

$f (\frac{3}{4}) = 0.5 (- \sin (\frac{π}{2})) + \frac{2}{3}$

Step 6.4.2.1.5

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

$f (\frac{3}{4}) = 0.5 (- 1 \cdot 1) + \frac{2}{3}$

Step 6.4.2.1.6

Multiply $0.5 (- 1 \cdot 1)$ .

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Step 6.4.2.1.6.1

Multiply $- 1$ by $1$ .

$f (\frac{3}{4}) = 0.5 \cdot - 1 + \frac{2}{3}$

Step 6.4.2.1.6.2

Multiply $0.5$ by $- 1$ .

$f (\frac{3}{4}) = - 0.5 + \frac{2}{3}$

Step 6.4.2.2

To write $- 0.5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{3}{4}) = - 0.5 \cdot \frac{3}{3} + \frac{2}{3}$

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $- 0.5$ and $\frac{3}{3}$ .

$f (\frac{3}{4}) = \frac{- 0.5 \cdot 3}{3} + \frac{2}{3}$

Step 6.4.2.3.2

Combine the numerators over the common denominator.

$f (\frac{3}{4}) = \frac{- 0.5 \cdot 3 + 2}{3}$

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply $- 0.5$ by $3$ .

$f (\frac{3}{4}) = \frac{- 1.5 + 2}{3}$

Step 6.4.2.4.2

Add $- 1.5$ and $2$ .

$f (\frac{3}{4}) = \frac{0.5}{3}$

Step 6.4.2.5

Divide $0.5$ by $3$ .

$f (\frac{3}{4}) = 0.1\overline{6}$

Step 6.4.2.6

The final answer is $0.1\overline{6}$ .

$0.1\overline{6}$

Step 6.5

Find the point at $x = 1$ .

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Step 6.5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 0.5 \sin (2 π (1)) + \frac{2}{3}$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Multiply $2$ by $1$ .

$f (1) = 0.5 \sin (2 π) + \frac{2}{3}$

Step 6.5.2.1.2

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

$f (1) = 0.5 \sin (0) + \frac{2}{3}$

Step 6.5.2.1.3

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

$f (1) = 0.5 \cdot 0 + \frac{2}{3}$

Step 6.5.2.1.4

Multiply $0.5$ by $0$ .

$f (1) = 0 + \frac{2}{3}$

Step 6.5.2.2

Add $0$ and $\frac{2}{3}$ .

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

Step 6.5.2.3

The final answer is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & \frac{2}{3} \\ \frac{1}{4} & 1.167 \\ \frac{1}{2} & \frac{2}{3} \\ \frac{3}{4} & 0.167 \\ 1 & \frac{2}{3} \end{array}$

Step 7

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

Amplitude: $0.5$

Period: $1$

Phase Shift: None

Vertical Shift: $\frac{2}{3}$

$\begin{array}{c} x & f (x) \\ 0 & \frac{2}{3} \\ \frac{1}{4} & 1.167 \\ \frac{1}{2} & \frac{2}{3} \\ \frac{3}{4} & 0.167 \\ 1 & \frac{2}{3} \end{array}$

Step 8