Trigonometry Examples

y = - \frac{3}{2} \cdot cos (\frac{3}{2} x)

$y = - \frac{3}{2} \cdot \cos (\frac{3}{2} x)$

Step 1

Use the form

$a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = - \frac{3}{2}$

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

$c = 0$

$d = 0$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$\frac{3}{2}$

Step 3

Find the period of

$- \frac{3 \cos (\frac{3 x}{2})}{2}$ .

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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:

$\frac{0}{\frac{3}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

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

Step 4.4

Multiply

$0$ by

$\frac{2}{3}$ .

Phase Shift:

$0$

Phase Shift:

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

$\frac{3}{2}$

Period:

$\frac{4 π}{3}$

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.

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Cancel the common factor of

$0$ and

$2$ .

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Step 6.1.2.1.1

Factor

$2$ out of

$3 (0)$ .

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

Step 6.1.2.1.2

Cancel the common factors.

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Step 6.1.2.1.2.1

Factor

$2$ out of

$2$ .

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

Step 6.1.2.1.2.2

Cancel the common factor.

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

Step 6.1.2.1.2.3

Rewrite the expression.

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

Step 6.1.2.1.2.4

Divide

$3 \cdot (0)$ by

$1$ .

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

Step 6.1.2.2

Simplify the numerator.

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Step 6.1.2.2.1

Multiply

$3$ by

$0$ .

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

Step 6.1.2.2.2

The exact value of

$\cos (0)$ is

$1$ .

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

Step 6.1.2.3

Multiply

$3$ by

$1$ .

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

Step 6.1.2.4

The final answer is

$- \frac{3}{2}$ .

$- \frac{3}{2}$

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.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Combine

$3$ and

$\frac{π}{3}$ .

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

Step 6.2.2.2

Reduce the expression by cancelling the common factors.

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Step 6.2.2.2.1

Reduce the expression

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

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Rewrite the expression.

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

Step 6.2.2.2.2

Divide

$π$ by

$1$ .

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

Step 6.2.2.3

The exact value of

$\cos (\frac{π}{2})$ is

$0$ .

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

Step 6.2.2.4

Multiply

$3$ by

$0$ .

$f (\frac{π}{3}) = - \frac{0}{2}$

Step 6.2.2.5

Divide

$0$ by

$2$ .

$f (\frac{π}{3}) = - 0$

Step 6.2.2.6

Multiply

$- 1$ by

$0$ .

$f (\frac{π}{3}) = 0$

Step 6.2.2.7

The final answer is

$0$ .

$0$

Step 6.3

Find the point at

$x = \frac{2 π}{3}$ .

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Step 6.3.1

Replace the variable

$x$ with

$\frac{2 π}{3}$ in the expression.

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Combine

$3$ and

$\frac{2 π}{3}$ .

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

Step 6.3.2.2

Multiply

$3$ by

$2$ .

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

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

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

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Step 6.3.2.3.1.1

Factor

$3$ out of

$6 π$ .

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

Step 6.3.2.3.1.2

Factor

$3$ out of

$3$ .

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

Step 6.3.2.3.1.3

Cancel the common factor.

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

Step 6.3.2.3.1.4

Rewrite the expression.

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

Step 6.3.2.3.2

Divide

$2 π$ by

$1$ .

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

Step 6.3.2.4

Simplify the numerator.

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Step 6.3.2.4.1

Cancel the common factor of

$2$ .

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Step 6.3.2.4.1.1

Cancel the common factor.

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

Step 6.3.2.4.1.2

Divide

$π$ by

$1$ .

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

Step 6.3.2.4.2

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

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

Step 6.3.2.4.3

The exact value of

$\cos (0)$ is

$1$ .

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

Step 6.3.2.4.4

Multiply

$- 1$ by

$1$ .

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

Step 6.3.2.5

Simplify the expression.

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Step 6.3.2.5.1

Multiply

$3$ by

$- 1$ .

$f (\frac{2 π}{3}) = - \frac{- 3}{2}$

Step 6.3.2.5.2

Move the negative in front of the fraction.

$f (\frac{2 π}{3}) = \frac{3}{2}$

Step 6.3.2.6

The final answer is

$\frac{3}{2}$ .

$\frac{3}{2}$

Step 6.4

Find the point at

$x = π$ .

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Step 6.4.1

Replace the variable

$x$ with

$π$ in the expression.

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

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

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

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

Step 6.4.2.1.2

The exact value of

$\cos (\frac{π}{2})$ is

$0$ .

$f (π) = - \frac{3 \cdot 0}{2}$

Step 6.4.2.2

Simplify the expression.

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Step 6.4.2.2.1

Multiply

$3$ by

$0$ .

$f (π) = - \frac{0}{2}$

Step 6.4.2.2.2

Divide

$0$ by

$2$ .

$f (π) = - 0$

Step 6.4.2.2.3

Multiply

$- 1$ by

$0$ .

$f (π) = 0$

Step 6.4.2.3

The final answer is

$0$ .

$0$

Step 6.5

Find the point at

$x = \frac{4 π}{3}$ .

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Step 6.5.1

Replace the variable

$x$ with

$\frac{4 π}{3}$ in the expression.

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Combine

$3$ and

$\frac{4 π}{3}$ .

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

Step 6.5.2.2

Multiply

$3$ by

$4$ .

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

Step 6.5.2.3

Reduce the expression by cancelling the common factors.

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Step 6.5.2.3.1

Reduce the expression

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

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Step 6.5.2.3.1.1

Factor

$3$ out of

$12 π$ .

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

Step 6.5.2.3.1.2

Factor

$3$ out of

$3$ .

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

Step 6.5.2.3.1.3

Cancel the common factor.

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

Step 6.5.2.3.1.4

Rewrite the expression.

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

Step 6.5.2.3.2

Divide

$4 π$ by

$1$ .

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

Step 6.5.2.4

Simplify the numerator.

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Step 6.5.2.4.1

Cancel the common factor of

$4$ and

$2$ .

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Step 6.5.2.4.1.1

Factor

$2$ out of

$4 π$ .

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

Step 6.5.2.4.1.2

Cancel the common factors.

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Step 6.5.2.4.1.2.1

Factor

$2$ out of

$2$ .

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

Step 6.5.2.4.1.2.2

Cancel the common factor.

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

Step 6.5.2.4.1.2.3

Rewrite the expression.

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

Step 6.5.2.4.1.2.4

Divide

$2 π$ by

$1$ .

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

Step 6.5.2.4.2

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$f (\frac{4 π}{3}) = - \frac{3 \cos (0)}{2}$

Step 6.5.2.4.3

The exact value of

$\cos (0)$ is

$1$ .

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

Step 6.5.2.5

Multiply

$3$ by

$1$ .

$f (\frac{4 π}{3}) = - \frac{3}{2}$

Step 6.5.2.6

The final answer is

$- \frac{3}{2}$ .

$- \frac{3}{2}$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude:

$\frac{3}{2}$

Period:

$\frac{4 π}{3}$

Phase Shift: None

Vertical Shift: None

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

Step 8