Trigonometry Examples

$y = - \sin (\frac{2 π x}{3})$

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.

$a = - 1$

$b = \frac{2 π}{3}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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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 $\frac{2 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{3} |}$

Step 3.3

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Step 3.5

Cancel the common factor of $2 π$ .

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Step 3.5.1

Cancel the common factor.

$2 π \frac{3}{2 π}$

Step 3.5.2

Rewrite the expression.

$3$

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: $\frac{0}{\frac{2 π}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{2 π})$

Step 4.4

Multiply $0$ by $\frac{3}{2 π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $3$

Phase Shift: None

Vertical Shift: None

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.

$f (0) = - \sin (\frac{2 π (0)}{3})$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Cancel the common factor of $0$ and $3$ .

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Step 6.1.2.1.1

Factor $3$ out of $2 π (0)$ .

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

Step 6.1.2.1.2

Cancel the common factors.

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Step 6.1.2.1.2.1

Factor $3$ out of $3$ .

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

Step 6.1.2.1.2.2

Cancel the common factor.

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

Step 6.1.2.1.2.3

Rewrite the expression.

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

Step 6.1.2.1.2.4

Divide $2 π \cdot (0)$ by $1$ .

$f (0) = - \sin (2 π \cdot (0))$

Step 6.1.2.2

Multiply $2 π (0)$ .

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Step 6.1.2.2.1

Multiply $0$ by $2$ .

$f (0) = - \sin (0 π)$

Step 6.1.2.2.2

Multiply $0$ by $π$ .

$f (0) = - \sin (0)$

Step 6.1.2.3

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

$f (0) = - 0$

Step 6.1.2.4

Multiply $- 1$ by $0$ .

$f (0) = 0$

Step 6.1.2.5

The final answer is $0$ .

$0$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify the numerator.

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Step 6.2.2.1.1

Combine $\frac{3}{4}$ and $2$ .

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

Step 6.2.2.1.2

Combine $\frac{3 \cdot 2}{4}$ and $π$ .

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

Step 6.2.2.2

Reduce the expression by cancelling the common factors.

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Step 6.2.2.2.1

Multiply $3$ by $2$ .

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

Step 6.2.2.2.2

Reduce the expression $\frac{6 π}{4}$ by cancelling the common factors.

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Step 6.2.2.2.2.1

Factor $2$ out of $6 π$ .

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

Step 6.2.2.2.2.2

Factor $2$ out of $4$ .

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

Step 6.2.2.2.2.3

Cancel the common factor.

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

Step 6.2.2.2.2.4

Rewrite the expression.

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

Step 6.2.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2.2.4

Cancel the common factor of $3$ .

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Step 6.2.2.4.1

Factor $3$ out of $3 π$ .

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

Step 6.2.2.4.2

Cancel the common factor.

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

Step 6.2.2.4.3

Rewrite the expression.

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

Step 6.2.2.5

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

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

Step 6.2.2.6

Multiply $- 1$ by $1$ .

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

Step 6.2.2.7

The final answer is $- 1$ .

$- 1$

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify the numerator.

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Step 6.3.2.1.1

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

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

Step 6.3.2.1.2

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

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

Step 6.3.2.2

Multiply $3$ by $2$ .

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

Step 6.3.2.3

Reduce the expression by cancelling the common factors.

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Step 6.3.2.3.1

Reduce the expression $\frac{6 π}{2}$ by cancelling the common factors.

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Step 6.3.2.3.1.1

Factor $2$ out of $6 π$ .

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

Step 6.3.2.3.1.2

Factor $2$ out of $2$ .

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

Step 6.3.2.3.1.3

Cancel the common factor.

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

Step 6.3.2.3.1.4

Rewrite the expression.

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

Step 6.3.2.3.2

Divide $3 π$ by $1$ .

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

Step 6.3.2.4

Cancel the common factor of $3$ .

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Step 6.3.2.4.1

Cancel the common factor.

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

Step 6.3.2.4.2

Divide $π$ by $1$ .

$f (\frac{3}{2}) = - \sin (π)$

Step 6.3.2.5

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

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

Step 6.3.2.6

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

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

Step 6.3.2.7

Multiply $- 1$ by $0$ .

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

Step 6.3.2.8

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{9}{4}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{9}{4}$ in the expression.

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify the numerator.

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Step 6.4.2.1.1

Combine $\frac{9}{4}$ and $2$ .

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

Step 6.4.2.1.2

Combine $\frac{9 \cdot 2}{4}$ and $π$ .

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

Step 6.4.2.2

Reduce the expression by cancelling the common factors.

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Step 6.4.2.2.1

Multiply $9$ by $2$ .

$f (\frac{9}{4}) = - \sin (\frac{\frac{18 π}{4}}{3})$

Step 6.4.2.2.2

Reduce the expression $\frac{18 π}{4}$ by cancelling the common factors.

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Step 6.4.2.2.2.1

Factor $2$ out of $18 π$ .

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

Step 6.4.2.2.2.2

Factor $2$ out of $4$ .

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

Step 6.4.2.2.2.3

Cancel the common factor.

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

Step 6.4.2.2.2.4

Rewrite the expression.

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

Step 6.4.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.4.2.4

Cancel the common factor of $3$ .

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Step 6.4.2.4.1

Factor $3$ out of $9 π$ .

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

Step 6.4.2.4.2

Cancel the common factor.

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

Step 6.4.2.4.3

Rewrite the expression.

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

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.

$f (\frac{9}{4}) = \sin (\frac{π}{2})$

Step 6.4.2.6

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

$f (\frac{9}{4}) = - (- 1 \cdot 1)$

Step 6.4.2.7

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

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Step 6.4.2.7.1

Multiply $- 1$ by $1$ .

$f (\frac{9}{4}) = 1$

Step 6.4.2.7.2

Multiply $- 1$ by $- 1$ .

$f (\frac{9}{4}) = 1$

Step 6.4.2.8

The final answer is $1$ .

$1$

Step 6.5

Find the point at $x = 3$ .

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Step 6.5.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = - \sin (\frac{2 π (3)}{3})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Divide $2 π$ by $1$ .

$f (3) = - \sin (2 π)$

Step 6.5.2.2

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

$f (3) = - \sin (0)$

Step 6.5.2.3

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

$f (3) = - 0$

Step 6.5.2.4

Multiply $- 1$ by $0$ .

$f (3) = 0$

Step 6.5.2.5

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{3}{4} & - 1 \\ \frac{3}{2} & 0 \\ \frac{9}{4} & 1 \\ 3 & 0 \end{array}$

Step 7

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

Amplitude: $1$

Period: $3$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{3}{4} & - 1 \\ \frac{3}{2} & 0 \\ \frac{9}{4} & 1 \\ 3 & 0 \end{array}$

Step 8