Trigonometry Examples

$y = \frac{3}{4} \cdot \sin (2 x) + 3$

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.

$a = \frac{3}{4}$

$b = 2$

$c = 0$

$d = 3$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{3}{4}$

Step 3

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

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Step 3.1

Find the period of $\frac{3 \sin (2 x)}{4}$ .

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.1.4.2

Divide $π$ by $1$ .

$π$

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.4.2

Divide $π$ by $1$ .

$π$

Step 3.3

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

$π$

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: $\frac{0}{2}$

Step 4.3

Divide $0$ by $2$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{3}{4}$

Period: $π$

Phase Shift: None

Vertical Shift: $3$

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.

$f (0) = \frac{3 \sin (2 (0))}{4} + 3$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $2$ by $0$ .

$f (0) = \frac{3 \sin (0)}{4} + 3$

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 $0$ .

$f (0) = \frac{3 \cdot 0}{4} + 3$

Step 6.1.2.2.2

Multiply $3$ by $0$ .

$f (0) = \frac{0}{4} + 3$

Step 6.1.2.2.3

Divide $0$ by $4$ .

$f (0) = 0 + 3$

Step 6.1.2.3

Add $0$ and $3$ .

$f (0) = 3$

Step 6.1.2.4

The final answer is $3$ .

$3$

Step 6.2

Find the point at $x = \frac{π}{4}$ .

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Step 6.2.1

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

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

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

Simplify the numerator.

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Step 6.2.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.1.1

Factor $2$ out of $4$ .

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

Step 6.2.2.1.1.1.2

Cancel the common factor.

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

Step 6.2.2.1.1.1.3

Rewrite the expression.

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

Step 6.2.2.1.1.2

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

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

Step 6.2.2.1.2

Multiply $3$ by $1$ .

$f (\frac{π}{4}) = \frac{3}{4} + 3$

Step 6.2.2.2

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

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

Step 6.2.2.3

Combine $3$ and $\frac{4}{4}$ .

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

Step 6.2.2.4

Combine the numerators over the common denominator.

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

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Multiply $3$ by $4$ .

$f (\frac{π}{4}) = \frac{3 + 12}{4}$

Step 6.2.2.5.2

Add $3$ and $12$ .

$f (\frac{π}{4}) = \frac{15}{4}$

Step 6.2.2.6

The final answer is $\frac{15}{4}$ .

$\frac{15}{4}$

Step 6.3

Find the point at $x = \frac{π}{2}$ .

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Step 6.3.1

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

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

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

Simplify the numerator.

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Step 6.3.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.1.1

Cancel the common factor.

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

Step 6.3.2.1.1.1.2

Rewrite the expression.

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

Step 6.3.2.1.1.2

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

$f (\frac{π}{2}) = \frac{3 \sin (0)}{4} + 3$

Step 6.3.2.1.1.3

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

$f (\frac{π}{2}) = \frac{3 \cdot 0}{4} + 3$

Step 6.3.2.1.2

Multiply $3$ by $0$ .

$f (\frac{π}{2}) = \frac{0}{4} + 3$

Step 6.3.2.1.3

Divide $0$ by $4$ .

$f (\frac{π}{2}) = 0 + 3$

Step 6.3.2.2

Add $0$ and $3$ .

$f (\frac{π}{2}) = 3$

Step 6.3.2.3

The final answer is $3$ .

$3$

Step 6.4

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

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Step 6.4.1

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

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

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

Simplify the numerator.

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Step 6.4.2.1.1.1

Cancel the common factor of $2$ .

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Step 6.4.2.1.1.1.1

Factor $2$ out of $4$ .

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

Step 6.4.2.1.1.1.2

Cancel the common factor.

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

Step 6.4.2.1.1.1.3

Rewrite the expression.

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

Step 6.4.2.1.1.2

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.

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

Step 6.4.2.1.1.3

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

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

Step 6.4.2.1.1.4

Multiply $- 1$ by $1$ .

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

Step 6.4.2.1.2

Multiply $3$ by $- 1$ .

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

Step 6.4.2.1.3

Move the negative in front of the fraction.

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

Step 6.4.2.2

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

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

Step 6.4.2.3

Combine $3$ and $\frac{4}{4}$ .

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

Step 6.4.2.4

Combine the numerators over the common denominator.

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

Step 6.4.2.5

Simplify the numerator.

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Step 6.4.2.5.1

Multiply $3$ by $4$ .

$f (\frac{3 π}{4}) = \frac{- 3 + 12}{4}$

Step 6.4.2.5.2

Add $- 3$ and $12$ .

$f (\frac{3 π}{4}) = \frac{9}{4}$

Step 6.4.2.6

The final answer is $\frac{9}{4}$ .

$\frac{9}{4}$

Step 6.5

Find the point at $x = π$ .

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Step 6.5.1

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

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

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

Simplify the numerator.

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Step 6.5.2.1.1.1

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

$f (π) = \frac{3 \sin (0)}{4} + 3$

Step 6.5.2.1.1.2

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

$f (π) = \frac{3 \cdot 0}{4} + 3$

Step 6.5.2.1.2

Multiply $3$ by $0$ .

$f (π) = \frac{0}{4} + 3$

Step 6.5.2.1.3

Divide $0$ by $4$ .

$f (π) = 0 + 3$

Step 6.5.2.2

Add $0$ and $3$ .

$f (π) = 3$

Step 6.5.2.3

The final answer is $3$ .

$3$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 3 \\ \frac{π}{4} & \frac{15}{4} \\ \frac{π}{2} & 3 \\ \frac{3 π}{4} & \frac{9}{4} \\ π & 3 \end{array}$

Step 7

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

Amplitude: $\frac{3}{4}$

Period: $π$

Phase Shift: None

Vertical Shift: $3$

$\begin{array}{c} x & f (x) \\ 0 & 3 \\ \frac{π}{4} & \frac{15}{4} \\ \frac{π}{2} & 3 \\ \frac{3 π}{4} & \frac{9}{4} \\ π & 3 \end{array}$

Step 8