Trigonometry Examples

$y = \cos (x - \frac{π}{3})$

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.

$a = 1$

$b = 1$

$c = \frac{π}{3}$

$d = 0$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$1$

Step 3

Find the period of

$\cos (x - \frac{π}{3})$ .

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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

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 3.4

Divide

$2 π$ by

$1$ .

$2 π$

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:

$\frac{\frac{π}{3}}{1}$

Step 4.3

Divide

$\frac{π}{3}$ by

$1$ .

Phase Shift:

$\frac{π}{3}$

Phase Shift:

$\frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$2 π$

Phase Shift:

$\frac{π}{3}$ (

$\frac{π}{3}$ 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

$x = \frac{π}{3}$ .

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Step 6.1.1

Replace the variable

$x$ with

$\frac{π}{3}$ in the expression.

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Combine the numerators over the common denominator.

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

Step 6.1.2.2

Subtract

$π$ from

$π$ .

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

Step 6.1.2.3

Divide

$0$ by

$3$ .

$f (\frac{π}{3}) = \cos (0)$

Step 6.1.2.4

The exact value of

$\cos (0)$ is

$1$ .

$f (\frac{π}{3}) = 1$

Step 6.1.2.5

The final answer is

$1$ .

$1$

Step 6.2

Find the point at

$x = \frac{5 π}{6}$ .

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Step 6.2.1

Replace the variable

$x$ with

$\frac{5 π}{6}$ in the expression.

$f (\frac{5 π}{6}) = \cos ((\frac{5 π}{6}) - \frac{π}{3})$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

To write

$- \frac{π}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (\frac{5 π}{6}) = \cos (\frac{5 π}{6} - \frac{π}{3} \cdot \frac{2}{2})$

Step 6.2.2.2

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 6.2.2.2.1

Multiply

$\frac{π}{3}$ by

$\frac{2}{2}$ .

$f (\frac{5 π}{6}) = \cos (\frac{5 π}{6} - \frac{π \cdot 2}{3 \cdot 2})$

Step 6.2.2.2.2

Multiply

$3$ by

$2$ .

$f (\frac{5 π}{6}) = \cos (\frac{5 π}{6} - \frac{π \cdot 2}{6})$

Step 6.2.2.3

Combine the numerators over the common denominator.

$f (\frac{5 π}{6}) = \cos (\frac{5 π - π \cdot 2}{6})$

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Multiply

$2$ by

$- 1$ .

$f (\frac{5 π}{6}) = \cos (\frac{5 π - 2 π}{6})$

Step 6.2.2.4.2

Subtract

$2 π$ from

$5 π$ .

$f (\frac{5 π}{6}) = \cos (\frac{3 π}{6})$

Step 6.2.2.5

Cancel the common factor of

$3$ and

$6$ .

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Step 6.2.2.5.1

Factor

$3$ out of

$3 π$ .

$f (\frac{5 π}{6}) = \cos (\frac{3 (π)}{6})$

Step 6.2.2.5.2

Cancel the common factors.

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Step 6.2.2.5.2.1

Factor

$3$ out of

$6$ .

$f (\frac{5 π}{6}) = \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.5.2.2

Cancel the common factor.

$f (\frac{5 π}{6}) = \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.5.2.3

Rewrite the expression.

$f (\frac{5 π}{6}) = \cos (\frac{π}{2})$

Step 6.2.2.6

The exact value of

$\cos (\frac{π}{2})$ is

$0$ .

$f (\frac{5 π}{6}) = 0$

Step 6.2.2.7

The final answer is

$0$ .

$0$

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.

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Combine the numerators over the common denominator.

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

Step 6.3.2.2

Subtract

$π$ from

$4 π$ .

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

Step 6.3.2.3

Cancel the common factor of

$3$ .

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Step 6.3.2.3.1

Cancel the common factor.

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

Step 6.3.2.3.2

Divide

$π$ by

$1$ .

$f (\frac{4 π}{3}) = \cos (π)$

Step 6.3.2.4

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.

$f (\frac{4 π}{3}) = - \cos (0)$

Step 6.3.2.5

The exact value of

$\cos (0)$ is

$1$ .

$f (\frac{4 π}{3}) = - 1 \cdot 1$

Step 6.3.2.6

Multiply

$- 1$ by

$1$ .

$f (\frac{4 π}{3}) = - 1$

Step 6.3.2.7

The final answer is

$- 1$ .

$- 1$

Step 6.4

Find the point at

$x = \frac{11 π}{6}$ .

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Step 6.4.1

Replace the variable

$x$ with

$\frac{11 π}{6}$ in the expression.

$f (\frac{11 π}{6}) = \cos ((\frac{11 π}{6}) - \frac{π}{3})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

To write

$- \frac{π}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (\frac{11 π}{6}) = \cos (\frac{11 π}{6} - \frac{π}{3} \cdot \frac{2}{2})$

Step 6.4.2.2

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 6.4.2.2.1

Multiply

$\frac{π}{3}$ by

$\frac{2}{2}$ .

$f (\frac{11 π}{6}) = \cos (\frac{11 π}{6} - \frac{π \cdot 2}{3 \cdot 2})$

Step 6.4.2.2.2

Multiply

$3$ by

$2$ .

$f (\frac{11 π}{6}) = \cos (\frac{11 π}{6} - \frac{π \cdot 2}{6})$

Step 6.4.2.3

Combine the numerators over the common denominator.

$f (\frac{11 π}{6}) = \cos (\frac{11 π - π \cdot 2}{6})$

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply

$2$ by

$- 1$ .

$f (\frac{11 π}{6}) = \cos (\frac{11 π - 2 π}{6})$

Step 6.4.2.4.2

Subtract

$2 π$ from

$11 π$ .

$f (\frac{11 π}{6}) = \cos (\frac{9 π}{6})$

Step 6.4.2.5

Cancel the common factor of

$9$ and

$6$ .

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Step 6.4.2.5.1

Factor

$3$ out of

$9 π$ .

$f (\frac{11 π}{6}) = \cos (\frac{3 (3 π)}{6})$

Step 6.4.2.5.2

Cancel the common factors.

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Step 6.4.2.5.2.1

Factor

$3$ out of

$6$ .

$f (\frac{11 π}{6}) = \cos (\frac{3 (3 π)}{3 (2)})$

Step 6.4.2.5.2.2

Cancel the common factor.

$f (\frac{11 π}{6}) = \cos (\frac{3 (3 π)}{3 \cdot 2})$

Step 6.4.2.5.2.3

Rewrite the expression.

$f (\frac{11 π}{6}) = \cos (\frac{3 π}{2})$

Step 6.4.2.6

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

$f (\frac{11 π}{6}) = \cos (\frac{π}{2})$

Step 6.4.2.7

The exact value of

$\cos (\frac{π}{2})$ is

$0$ .

$f (\frac{11 π}{6}) = 0$

Step 6.4.2.8

The final answer is

$0$ .

$0$

Step 6.5

Find the point at

$x = \frac{7 π}{3}$ .

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Step 6.5.1

Replace the variable

$x$ with

$\frac{7 π}{3}$ in the expression.

$f (\frac{7 π}{3}) = \cos ((\frac{7 π}{3}) - \frac{π}{3})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Combine the numerators over the common denominator.

$f (\frac{7 π}{3}) = \cos (\frac{7 π - π}{3})$

Step 6.5.2.2

Subtract

$π$ from

$7 π$ .

$f (\frac{7 π}{3}) = \cos (\frac{6 π}{3})$

Step 6.5.2.3

Cancel the common factor of

$6$ and

$3$ .

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Step 6.5.2.3.1

Factor

$3$ out of

$6 π$ .

$f (\frac{7 π}{3}) = \cos (\frac{3 (2 π)}{3})$

Step 6.5.2.3.2

Cancel the common factors.

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Step 6.5.2.3.2.1

Factor

$3$ out of

$3$ .

$f (\frac{7 π}{3}) = \cos (\frac{3 (2 π)}{3 (1)})$

Step 6.5.2.3.2.2

Cancel the common factor.

$f (\frac{7 π}{3}) = \cos (\frac{3 (2 π)}{3 \cdot 1})$

Step 6.5.2.3.2.3

Rewrite the expression.

$f (\frac{7 π}{3}) = \cos (\frac{2 π}{1})$

Step 6.5.2.3.2.4

Divide

$2 π$ by

$1$ .

$f (\frac{7 π}{3}) = \cos (2 π)$

Step 6.5.2.4

Subtract full rotations of

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

$0$ and less than

$2 π$ .

$f (\frac{7 π}{3}) = \cos (0)$

Step 6.5.2.5

The exact value of

$\cos (0)$ is

$1$ .

$f (\frac{7 π}{3}) = 1$

Step 6.5.2.6

The final answer is

$1$ .

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{3} & 1 \\ \frac{5 π}{6} & 0 \\ \frac{4 π}{3} & - 1 \\ \frac{11 π}{6} & 0 \\ \frac{7 π}{3} & 1 \end{array}$

Step 7

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

Amplitude:

$1$

Period:

$2 π$

Phase Shift:

$\frac{π}{3}$ (

$\frac{π}{3}$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{π}{3} & 1 \\ \frac{5 π}{6} & 0 \\ \frac{4 π}{3} & - 1 \\ \frac{11 π}{6} & 0 \\ \frac{7 π}{3} & 1 \end{array}$

Step 8