Trigonometry Examples

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

Step 1

Rewrite the expression as $- 2 \cos (\frac{π x}{2} - \frac{π}{4}) + 5$ .

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

Step 2

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

$a = - 2$

$b = \frac{π}{2}$

$c = \frac{π}{4}$

$d = 5$

Step 3

Find the amplitude $| a |$ .

Amplitude: $2$

Step 4

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 4.1

Find the period of $- 2 \cos (\frac{π x}{2} - \frac{π}{4})$ .

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Step 4.1.1

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

$\frac{2 π}{| b |}$

Step 4.1.2

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

$\frac{2 π}{| \frac{π}{2} |}$

Step 4.1.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 4.1.5

Cancel the common factor of $π$ .

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Step 4.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 4.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 4.1.5.3

Rewrite the expression.

$2 \cdot 2$

Step 4.1.6

Multiply $2$ by $2$ .

$4$

Step 4.2

Find the period of $5$ .

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Step 4.2.1

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

$\frac{2 π}{| b |}$

Step 4.2.2

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

$\frac{2 π}{| \frac{π}{2} |}$

Step 4.2.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 4.2.5

Cancel the common factor of $π$ .

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Step 4.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 4.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 4.2.5.3

Rewrite the expression.

$2 \cdot 2$

Step 4.2.6

Multiply $2$ by $2$ .

$4$

Step 4.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$4$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{\frac{π}{4}}{\frac{π}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{4} \cdot \frac{2}{π}$

Step 5.4

Cancel the common factor of $π$ .

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Step 5.4.1

Cancel the common factor.

Phase Shift: $\frac{π}{4} \cdot \frac{2}{π}$

Step 5.4.2

Rewrite the expression.

Phase Shift: $\frac{1}{4} \cdot 2$

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Factor $2$ out of $4$ .

Phase Shift: $\frac{1}{2 (2)} \cdot 2$

Step 5.5.2

Cancel the common factor.

Phase Shift: $\frac{1}{2 \cdot 2} \cdot 2$

Step 5.5.3

Rewrite the expression.

Phase Shift: $\frac{1}{2}$

Step 6

List the properties of the trigonometric function.

Amplitude: $2$

Period: $4$

Phase Shift: $\frac{1}{2}$ ( $\frac{1}{2}$ to the right)

Vertical Shift: $5$

Step 7