Trigonometry Examples

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

Step 1

Rewrite the expression as $\cos (\frac{x}{2} - \frac{π}{2}) + 1$ .

$\cos (\frac{x}{2} - \frac{π}{2}) + 1$

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.

$a = 1$

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

$c = \frac{π}{2}$

$d = 1$

Step 3

Find the amplitude $| a |$ .

Amplitude: $1$

Step 4

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

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Step 4.1

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

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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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 4.1.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 4.1.5

Multiply $2$ by $2$ .

$4 π$

Step 4.2

Find the period of $1$ .

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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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 4.2.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 4.2.5

Multiply $2$ by $2$ .

$4 π$

Step 4.3

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

$4 π$

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: $\frac{\frac{π}{2}}{\frac{1}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot 2$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

Phase Shift: $\frac{π}{2} \cdot 2$

Step 5.4.2

Rewrite the expression.

Phase Shift: $π$

Step 6

List the properties of the trigonometric function.

Amplitude: $1$

Period: $4 π$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: $1$

Step 7