Trigonometry Examples

y = \frac{1}{2} \cos (\frac{4 π x}{5} - 5 π)

$y = \frac{1}{2} \cos (\frac{4 π x}{5} - 5 π)$

Step 1

Use the form

a \cos (b x - c) + d

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

a = \frac{1}{2}

$a = \frac{1}{2}$

b = \frac{4 π}{5}

$b = \frac{4 π}{5}$

c = 5 π

$c = 5 π$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

\frac{1}{2}

$\frac{1}{2}$

Step 3

Find the period of

\frac{\cos (\frac{4 π x}{5} - 5 π)}{2}

$\frac{\cos (\frac{4 π x}{5} - 5 π)}{2}$ .

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Step 3.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2

Replace

b

$b$ with

\frac{4 π}{5}

$\frac{4 π}{5}$ in the formula for period.

\frac{2 π}{| \frac{4 π}{5} |}

$\frac{2 π}{| \frac{4 π}{5} |}$

Step 3.3

\frac{4 π}{5}

$\frac{4 π}{5}$ is approximately

2.51327412

$2.51327412$ which is positive so remove the absolute value

\frac{2 π}{\frac{4 π}{5}}

$\frac{2 π}{\frac{4 π}{5}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

2 π \frac{5}{4 π}

$2 π \frac{5}{4 π}$

Step 3.5

Cancel the common factor of

2 π

$2 π$ .

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Step 3.5.1

Factor

2 π

$2 π$ out of

4 π

$4 π$ .

2 π \frac{5}{2 π (2)}

$2 π \frac{5}{2 π (2)}$

Step 3.5.2

Cancel the common factor.

2 π \frac{5}{2 π \cdot 2}

$2 π \frac{5}{2 π \cdot 2}$

Step 3.5.3

Rewrite the expression.

\frac{5}{2}

$\frac{5}{2}$

\frac{5}{2}

$\frac{5}{2}$

\frac{5}{2}

$\frac{5}{2}$

Step 4

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 4.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{5 π}{\frac{4 π}{5}}

$\frac{5 π}{\frac{4 π}{5}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

5 π (\frac{5}{4 π})

$5 π (\frac{5}{4 π})$

Step 4.4

Cancel the common factor of

π

$π$ .

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Step 4.4.1

Factor

π

$π$ out of

5 π

$5 π$ .

Phase Shift:

π \cdot (5 (\frac{5}{4 π}))

$π \cdot (5 (\frac{5}{4 π}))$

Step 4.4.2

Factor

π

$π$ out of

4 π

$4 π$ .

Phase Shift:

π \cdot (5 (\frac{5}{π \cdot 4}))

$π \cdot (5 (\frac{5}{π \cdot 4}))$

Step 4.4.3

Cancel the common factor.

Phase Shift:

π \cdot (5 (\frac{5}{π \cdot 4}))

$π \cdot (5 (\frac{5}{π \cdot 4}))$

Step 4.4.4

Rewrite the expression.

Phase Shift:

5 (\frac{5}{4})

$5 (\frac{5}{4})$

Phase Shift:

5 (\frac{5}{4})

$5 (\frac{5}{4})$

Step 4.5

Combine

5

$5$ and

\frac{5}{4}

$\frac{5}{4}$ .

Phase Shift:

\frac{5 \cdot 5}{4}

$\frac{5 \cdot 5}{4}$

Step 4.6

Multiply

5

$5$ by

5

$5$ .

Phase Shift:

\frac{25}{4}

$\frac{25}{4}$

Phase Shift:

\frac{25}{4}

$\frac{25}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude:

\frac{1}{2}

$\frac{1}{2}$

Period:

\frac{5}{2}

$\frac{5}{2}$

Phase Shift:

\frac{25}{4}

$\frac{25}{4}$ (

\frac{25}{4}

$\frac{25}{4}$ to the right)

Vertical Shift: None

Step 6