Trigonometry Examples

$h (t) = 40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55$

Step 1

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

$a = 40$

$b = \frac{5 π}{4}$

$c = \frac{π}{2}$

$d = 55$

Step 2

Find the amplitude $| a |$ .

Amplitude: $40$

Step 3

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

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Step 3.1

Find the period of $40 \sin (\frac{5 π x}{4} - \frac{π}{2})$ .

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

$\frac{2 π}{| \frac{5 π}{4} |}$

Step 3.1.3

$\frac{5 π}{4}$ is approximately $3.92699081$ which is positive so remove the absolute value

$\frac{2 π}{\frac{5 π}{4}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{5 π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{5 π}$

Step 3.1.5.2

Factor $π$ out of $5 π$ .

$π \cdot 2 \frac{4}{π \cdot 5}$

Step 3.1.5.3

Cancel the common factor.

$π \cdot 2 \frac{4}{π \cdot 5}$

Step 3.1.5.4

Rewrite the expression.

$2 (\frac{4}{5})$

Step 3.1.6

Combine $2$ and $\frac{4}{5}$ .

$\frac{2 \cdot 4}{5}$

Step 3.1.7

Multiply $2$ by $4$ .

$\frac{8}{5}$

Step 3.2

Find the period of $55$ .

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

$\frac{2 π}{| \frac{5 π}{4} |}$

Step 3.2.3

$\frac{5 π}{4}$ is approximately $3.92699081$ which is positive so remove the absolute value

$\frac{2 π}{\frac{5 π}{4}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{5 π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{5 π}$

Step 3.2.5.2

Factor $π$ out of $5 π$ .

$π \cdot 2 \frac{4}{π \cdot 5}$

Step 3.2.5.3

Cancel the common factor.

$π \cdot 2 \frac{4}{π \cdot 5}$

Step 3.2.5.4

Rewrite the expression.

$2 (\frac{4}{5})$

Step 3.2.6

Combine $2$ and $\frac{4}{5}$ .

$\frac{2 \cdot 4}{5}$

Step 3.2.7

Multiply $2$ by $4$ .

$\frac{8}{5}$

Step 3.3

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

$\frac{8}{5}$

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: $\frac{\frac{π}{2}}{\frac{5 π}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{4}{5 π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Factor $π$ out of $5 π$ .

Phase Shift: $\frac{π}{2} \cdot \frac{4}{π \cdot 5}$

Step 4.4.2

Cancel the common factor.

Phase Shift: $\frac{π}{2} \cdot \frac{4}{π \cdot 5}$

Step 4.4.3

Rewrite the expression.

Phase Shift: $\frac{1}{2} \cdot \frac{4}{5}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Factor $2$ out of $4$ .

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

Step 4.5.2

Cancel the common factor.

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

Step 4.5.3

Rewrite the expression.

Phase Shift: $\frac{2}{5}$

Step 5

List the properties of the trigonometric function.

Amplitude: $40$

Period: $\frac{8}{5}$

Phase Shift: $\frac{2}{5}$ ( $\frac{2}{5}$ to the right)

Vertical Shift: $55$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = \frac{8}{5}$ .

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Step 6.1.1

Replace the variable $x$ with $\frac{8}{5}$ in the expression.

$f (\frac{8}{5}) = 40 \sin (\frac{5 π (\frac{8}{5})}{4} - \frac{π}{2}) + 55$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Simplify each term.

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Step 6.1.2.1.1.1

Simplify the numerator.

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Step 6.1.2.1.1.1.1

Combine $\frac{8}{5}$ and $5$ .

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{8 \cdot 5}{5} \cdot π}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.1.2

Combine $\frac{8 \cdot 5}{5}$ and $π$ .

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{8 \cdot (5 π)}{5}}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.2

Multiply $8$ by $5$ .

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{40 π}{5}}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.3

Reduce the expression by cancelling the common factors.

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Step 6.1.2.1.1.3.1

Reduce the expression $\frac{40 π}{5}$ by cancelling the common factors.

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Step 6.1.2.1.1.3.1.1

Factor $5$ out of $40 π$ .

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{5 (8 π)}{5}}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.3.1.2

Factor $5$ out of $5$ .

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{5 (8 π)}{5 (1)}}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.3.1.3

Cancel the common factor.

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{5 (8 π)}{5 \cdot 1}}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.3.1.4

Rewrite the expression.

$f (\frac{8}{5}) = 40 \sin (\frac{\frac{8 π}{1}}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.3.2

Divide $8 π$ by $1$ .

$f (\frac{8}{5}) = 40 \sin (\frac{8 π}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.4

Cancel the common factor of $8$ and $4$ .

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Step 6.1.2.1.1.4.1

Factor $4$ out of $8 π$ .

$f (\frac{8}{5}) = 40 \sin (\frac{4 (2 π)}{4} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.4.2

Cancel the common factors.

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Step 6.1.2.1.1.4.2.1

Factor $4$ out of $4$ .

$f (\frac{8}{5}) = 40 \sin (\frac{4 (2 π)}{4 (1)} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.4.2.2

Cancel the common factor.

$f (\frac{8}{5}) = 40 \sin (\frac{4 (2 π)}{4 \cdot 1} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.4.2.3

Rewrite the expression.

$f (\frac{8}{5}) = 40 \sin (\frac{2 π}{1} - \frac{π}{2}) + 55$

Step 6.1.2.1.1.4.2.4

Divide $2 π$ by $1$ .

$f (\frac{8}{5}) = 40 \sin (2 π - \frac{π}{2}) + 55$

Step 6.1.2.1.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{8}{5}) = 40 \sin (2 π \cdot \frac{2}{2} - \frac{π}{2}) + 55$

Step 6.1.2.1.3

Combine $2 π$ and $\frac{2}{2}$ .

$f (\frac{8}{5}) = 40 \sin (\frac{2 π \cdot 2}{2} - \frac{π}{2}) + 55$

Step 6.1.2.1.4

Combine the numerators over the common denominator.

$f (\frac{8}{5}) = 40 \sin (\frac{2 π \cdot 2 - π}{2}) + 55$

Step 6.1.2.1.5

Simplify the numerator.

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Step 6.1.2.1.5.1

Multiply $2$ by $2$ .

$f (\frac{8}{5}) = 40 \sin (\frac{4 π - π}{2}) + 55$

Step 6.1.2.1.5.2

Subtract $π$ from $4 π$ .

$f (\frac{8}{5}) = 40 \sin (\frac{3 π}{2}) + 55$

Step 6.1.2.1.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{8}{5}) = 40 (- \sin (\frac{π}{2})) + 55$

Step 6.1.2.1.7

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{8}{5}) = 40 (- 1 \cdot 1) + 55$

Step 6.1.2.1.8

Multiply $40 (- 1 \cdot 1)$ .

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Step 6.1.2.1.8.1

Multiply $- 1$ by $1$ .

$f (\frac{8}{5}) = 40 \cdot - 1 + 55$

Step 6.1.2.1.8.2

Multiply $40$ by $- 1$ .

$f (\frac{8}{5}) = - 40 + 55$

Step 6.1.2.2

Add $- 40$ and $55$ .

$f (\frac{8}{5}) = 15$

Step 6.1.2.3

The final answer is $15$ .

$15$

Step 6.2

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{8}{5} & 15 \\ \frac{16}{5} & 15 \\ \frac{24}{5} & 15 \\ \frac{32}{5} & 15 \\ 8 & 15 \end{array}$

Step 7

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

Amplitude: $40$

Period: $\frac{8}{5}$

Phase Shift: $\frac{2}{5}$ ( $\frac{2}{5}$ to the right)

Vertical Shift: $55$

$\begin{array}{c} x & f (x) \\ \frac{8}{5} & 15 \\ \frac{16}{5} & 15 \\ \frac{24}{5} & 15 \\ \frac{32}{5} & 15 \\ 8 & 15 \end{array}$

Step 8