Trigonometry Examples

$f (t) = 50 \sin (\frac{π}{10} t - \frac{π}{2}) + 53$

Step 1

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

$a = 50$

$b = \frac{π}{10}$

$c = \frac{π}{2}$

$d = 53$

Step 2

Find the amplitude $| a |$ .

Amplitude: $50$

Step 3

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

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Step 3.1

Find the period of $50 \sin (\frac{π t}{10} - \frac{π}{2})$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{π}{10}$ in the formula for period.

$\frac{2 π}{| \frac{π}{10} |}$

Step 3.1.3

$\frac{π}{10}$ is approximately $0.31415926$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{10}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{10}{π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{10}{π}$

Step 3.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{10}{π}$

Step 3.1.5.3

Rewrite the expression.

$2 \cdot 10$

Step 3.1.6

Multiply $2$ by $10$ .

$20$

Step 3.2

Find the period of $53$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{π}{10}$ in the formula for period.

$\frac{2 π}{| \frac{π}{10} |}$

Step 3.2.3

$\frac{π}{10}$ is approximately $0.31415926$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{10}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{10}{π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{10}{π}$

Step 3.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{10}{π}$

Step 3.2.5.3

Rewrite the expression.

$2 \cdot 10$

Step 3.2.6

Multiply $2$ by $10$ .

$20$

Step 3.3

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

$20$

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: $\frac{\frac{π}{2}}{\frac{π}{10}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{10}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Cancel the common factor.

Phase Shift: $\frac{π}{2} \cdot \frac{10}{π}$

Step 4.4.2

Rewrite the expression.

Phase Shift: $\frac{1}{2} \cdot 10$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Factor $2$ out of $10$ .

Phase Shift: $\frac{1}{2} \cdot (2 (5))$

Step 4.5.2

Cancel the common factor.

Phase Shift: $\frac{1}{2} \cdot (2 \cdot 5)$

Step 4.5.3

Rewrite the expression.

Phase Shift: $5$

Step 5

List the properties of the trigonometric function.

Amplitude: $50$

Period: $20$

Phase Shift: $5$ ( $5$ to the right)

Vertical Shift: $53$

Step 6