Trigonometry Examples

$i = 15 \sin (100 π t + 0.6)$

Step 1

Subtract $15 \sin (100 π t + 0.6)$ from both sides of the equation.

$i - 15 \sin (100 π t + 0.6) = 0$

Step 2

Rewrite the expression as $- 15 \sin (100 π t + 0.6) + i$ .

$- 15 \sin (100 π t + 0.6) + i$

Step 3

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

$a = - 15$

$b = 100 π$

$c = - 0.6$

$d = i$

Step 4

Find the amplitude $| a |$ .

Amplitude: $15$

Step 5

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

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Step 5.1

Find the period of $- 15 \sin (100 π t + 0.6)$ .

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Step 5.1.1

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

$\frac{2 π}{| b |}$

Step 5.1.2

Replace $b$ with $100 π$ in the formula for period.

$\frac{2 π}{| 100 π |}$

Step 5.1.3

$100 π$ is approximately $314.15926535$ which is positive so remove the absolute value

$\frac{2 π}{100 π}$

Step 5.1.4

Cancel the common factor of $2$ and $100$ .

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Step 5.1.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{100 π}$

Step 5.1.4.2

Cancel the common factors.

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Step 5.1.4.2.1

Factor $2$ out of $100 π$ .

$\frac{2 (π)}{2 (50 π)}$

Step 5.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (50 π)}$

Step 5.1.4.2.3

Rewrite the expression.

$\frac{π}{50 π}$

Step 5.1.5

Cancel the common factor of $π$ .

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Step 5.1.5.1

Cancel the common factor.

$\frac{π}{50 π}$

Step 5.1.5.2

Rewrite the expression.

$\frac{1}{50}$

Step 5.2

Find the period of $i$ .

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Step 5.2.1

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

$\frac{2 π}{| b |}$

Step 5.2.2

Replace $b$ with $100 π$ in the formula for period.

$\frac{2 π}{| 100 π |}$

Step 5.2.3

$100 π$ is approximately $314.15926535$ which is positive so remove the absolute value

$\frac{2 π}{100 π}$

Step 5.2.4

Cancel the common factor of $2$ and $100$ .

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Step 5.2.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{100 π}$

Step 5.2.4.2

Cancel the common factors.

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Step 5.2.4.2.1

Factor $2$ out of $100 π$ .

$\frac{2 (π)}{2 (50 π)}$

Step 5.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (50 π)}$

Step 5.2.4.2.3

Rewrite the expression.

$\frac{π}{50 π}$

Step 5.2.5

Cancel the common factor of $π$ .

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Step 5.2.5.1

Cancel the common factor.

$\frac{π}{50 π}$

Step 5.2.5.2

Rewrite the expression.

$\frac{1}{50}$

Step 5.3

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

$\frac{1}{50}$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

Phase Shift: $\frac{- 0.6}{100 π}$

Step 6.3

Move the negative in front of the fraction.

Phase Shift: $- \frac{0.6}{100 π}$

Step 6.4

Replace $π$ with an approximation.

Phase Shift: $- \frac{0.6}{100 \cdot 3.14159265}$

Step 6.5

Multiply $100$ by $3.14159265$ .

Phase Shift: $- \frac{0.6}{314.15926535}$

Step 6.6

Divide $0.6$ by $314.15926535$ .

Phase Shift: $- 1 \cdot 0.00190985$

Step 6.7

Multiply $- 1$ by $0.00190985$ .

Phase Shift: $- 0.00190985$

Step 7

List the properties of the trigonometric function.

Amplitude: $15$

Period: $\frac{1}{50}$

Phase Shift: $- 0.00190985$ ( $0.00190985$ to the left)

Vertical Shift: $i$

Step 8