Trigonometry Examples

$y = - 3 \sin (2 π x + 4 π)$

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.

$a = - 3$

$b = 2 π$

$c = - 4 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $- 3 \sin (2 π x + 4 π)$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 3.3

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 3.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 3.5

Cancel the common factor of $π$ .

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Step 3.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 3.5.2

Rewrite the expression.

$1$

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: $\frac{- 4 π}{2 π}$

Step 4.3

Cancel the common factor of $- 4$ and $2$ .

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Step 4.3.1

Factor $2$ out of $- 4 π$ .

Phase Shift: $\frac{2 (- 2 π)}{2 π}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2 π$ .

Phase Shift: $\frac{2 (- 2 π)}{2 (π)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{2 (- 2 π)}{2 π}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{- 2 π}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Cancel the common factor.

Phase Shift: $\frac{- 2 π}{π}$

Step 4.4.2

Divide $- 2$ by $1$ .

Phase Shift: $- 2$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $1$

Phase Shift: $- 2$ ( $2$ to the left)

Vertical Shift: None

Step 6