Trigonometry Examples

$2 \sin (\frac{3 x}{2} - π)$

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.

$a = 2$

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

$c = π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

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

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Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{3}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3}{2} |}$

Step 3.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Step 3.5

Multiply $2 π \frac{2}{3}$ .

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Step 3.5.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Step 3.5.2

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Step 3.5.3

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

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: $\frac{π}{\frac{3}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{2}{3})$

Step 4.4

Combine $π$ and $\frac{2}{3}$ .

Phase Shift: $\frac{π \cdot 2}{3}$

Step 4.5

Move $2$ to the left of $π$ .

Phase Shift: $\frac{2 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{4 π}{3}$

Phase Shift: $\frac{2 π}{3}$ ( $\frac{2 π}{3}$ to the right)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = \frac{2 π}{3}$ .

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Step 6.1.1

Replace the variable $x$ with $\frac{2 π}{3}$ in the expression.

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

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

Combine $3$ and $\frac{2 π}{3}$ .

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

Step 6.1.2.1.2

Multiply $3$ by $2$ .

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

Step 6.1.2.1.3

Reduce the expression by cancelling the common factors.

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Step 6.1.2.1.3.1

Reduce the expression $\frac{6 π}{3}$ by cancelling the common factors.

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Step 6.1.2.1.3.1.1

Factor $3$ out of $6 π$ .

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

Step 6.1.2.1.3.1.2

Factor $3$ out of $3$ .

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

Step 6.1.2.1.3.1.3

Cancel the common factor.

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

Step 6.1.2.1.3.1.4

Rewrite the expression.

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

Step 6.1.2.1.3.2

Divide $2 π$ by $1$ .

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

Step 6.1.2.1.4

Cancel the common factor of $2$ .

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Step 6.1.2.1.4.1

Cancel the common factor.

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

Step 6.1.2.1.4.2

Divide $π$ by $1$ .

$f (\frac{2 π}{3}) = 2 \sin (π - π)$

Step 6.1.2.2

Subtract $π$ from $π$ .

$f (\frac{2 π}{3}) = 2 \sin (0)$

Step 6.1.2.3

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

$f (\frac{2 π}{3}) = 2 \cdot 0$

Step 6.1.2.4

Multiply $2$ by $0$ .

$f (\frac{2 π}{3}) = 0$

Step 6.1.2.5

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = π$ .

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Step 6.2.1

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

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

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

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

Step 6.2.2.2

Combine fractions.

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Step 6.2.2.2.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.2.2.2.2

Combine the numerators over the common denominator.

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

Step 6.2.2.3

Simplify the numerator.

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Step 6.2.2.3.1

Multiply $2$ by $- 1$ .

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

Step 6.2.2.3.2

Subtract $2 π$ from $3 π$ .

$f (π) = 2 \sin (\frac{π}{2})$

Step 6.2.2.4

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

$f (π) = 2 \cdot 1$

Step 6.2.2.5

Multiply $2$ by $1$ .

$f (π) = 2$

Step 6.2.2.6

The final answer is $2$ .

$2$

Step 6.3

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

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Step 6.3.1

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

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

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

Combine $3$ and $\frac{4 π}{3}$ .

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

Step 6.3.2.1.2

Multiply $3$ by $4$ .

$f (\frac{4 π}{3}) = 2 \sin (\frac{\frac{12 π}{3}}{2} - π)$

Step 6.3.2.1.3

Reduce the expression by cancelling the common factors.

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Step 6.3.2.1.3.1

Reduce the expression $\frac{12 π}{3}$ by cancelling the common factors.

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Step 6.3.2.1.3.1.1

Factor $3$ out of $12 π$ .

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

Step 6.3.2.1.3.1.2

Factor $3$ out of $3$ .

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

Step 6.3.2.1.3.1.3

Cancel the common factor.

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

Step 6.3.2.1.3.1.4

Rewrite the expression.

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

Step 6.3.2.1.3.2

Divide $4 π$ by $1$ .

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

Step 6.3.2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 6.3.2.1.4.1

Factor $2$ out of $4 π$ .

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

Step 6.3.2.1.4.2

Cancel the common factors.

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Step 6.3.2.1.4.2.1

Factor $2$ out of $2$ .

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

Step 6.3.2.1.4.2.2

Cancel the common factor.

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

Step 6.3.2.1.4.2.3

Rewrite the expression.

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

Step 6.3.2.1.4.2.4

Divide $2 π$ by $1$ .

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

Step 6.3.2.2

Subtract $π$ from $2 π$ .

$f (\frac{4 π}{3}) = 2 \sin (π)$

Step 6.3.2.3

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

$f (\frac{4 π}{3}) = 2 \sin (0)$

Step 6.3.2.4

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

$f (\frac{4 π}{3}) = 2 \cdot 0$

Step 6.3.2.5

Multiply $2$ by $0$ .

$f (\frac{4 π}{3}) = 0$

Step 6.3.2.6

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{5 π}{3}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{5 π}{3}$ in the expression.

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

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

Combine $3$ and $\frac{5 π}{3}$ .

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

Step 6.4.2.1.2

Multiply $3$ by $5$ .

$f (\frac{5 π}{3}) = 2 \sin (\frac{\frac{15 π}{3}}{2} - π)$

Step 6.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 6.4.2.1.3.1

Reduce the expression $\frac{15 π}{3}$ by cancelling the common factors.

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Step 6.4.2.1.3.1.1

Factor $3$ out of $15 π$ .

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

Step 6.4.2.1.3.1.2

Factor $3$ out of $3$ .

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

Step 6.4.2.1.3.1.3

Cancel the common factor.

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

Step 6.4.2.1.3.1.4

Rewrite the expression.

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

Step 6.4.2.1.3.2

Divide $5 π$ by $1$ .

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

Step 6.4.2.2

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

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

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.4.2.3.2

Combine the numerators over the common denominator.

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

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply $2$ by $- 1$ .

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

Step 6.4.2.4.2

Subtract $2 π$ from $5 π$ .

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

Step 6.4.2.5

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.

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

Step 6.4.2.6

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

$f (\frac{5 π}{3}) = 2 (- 1 \cdot 1)$

Step 6.4.2.7

Multiply $2 (- 1 \cdot 1)$ .

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Step 6.4.2.7.1

Multiply $- 1$ by $1$ .

$f (\frac{5 π}{3}) = 2 \cdot - 1$

Step 6.4.2.7.2

Multiply $2$ by $- 1$ .

$f (\frac{5 π}{3}) = - 2$

Step 6.4.2.8

The final answer is $- 2$ .

$- 2$

Step 6.5

Find the point at $x = 2 π$ .

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Step 6.5.1

Replace the variable $x$ with $2 π$ in the expression.

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of $2$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Divide $3 (π)$ by $1$ .

$f (2 π) = 2 \sin (3 (π) - π)$

$f (2 π) = 2 \sin (3 π - π)$

Step 6.5.2.2

Subtract $π$ from $3 π$ .

$f (2 π) = 2 \sin (2 π)$

Step 6.5.2.3

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

$f (2 π) = 2 \sin (0)$

Step 6.5.2.4

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

$f (2 π) = 2 \cdot 0$

Step 6.5.2.5

Multiply $2$ by $0$ .

$f (2 π) = 0$

Step 6.5.2.6

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{2 π}{3} & 0 \\ π & 2 \\ \frac{4 π}{3} & 0 \\ \frac{5 π}{3} & - 2 \\ 2 π & 0 \end{array}$

Step 7

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

Amplitude: $2$

Period: $\frac{4 π}{3}$

Phase Shift: $\frac{2 π}{3}$ ( $\frac{2 π}{3}$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{2 π}{3} & 0 \\ π & 2 \\ \frac{4 π}{3} & 0 \\ \frac{5 π}{3} & - 2 \\ 2 π & 0 \end{array}$

Step 8