Trigonometry Examples

$y = - 3 - 3 \sin (5 x + \frac{π}{4})$

Step 1

Rewrite the expression as $- 3 \sin (5 x + \frac{π}{4}) - 3$ .

$- 3 \sin (5 x + \frac{π}{4}) - 3$

Step 2

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

$a = - 3$

$b = 5$

$c = - \frac{π}{4}$

$d = - 3$

Step 3

Find the amplitude $| a |$ .

Amplitude: $3$

Step 4

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

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Step 4.1

Find the period of $- 3 \sin (5 x + \frac{π}{4})$ .

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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 $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 4.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 4.2

Find the period of $- 3$ .

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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 $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 4.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 4.3

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

$\frac{2 π}{5}$

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: $\frac{- \frac{π}{4}}{5}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4

Multiply $- \frac{π}{4} \cdot \frac{1}{5}$ .

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Step 5.4.1

Multiply $\frac{1}{5}$ by $\frac{π}{4}$ .

Phase Shift: $- \frac{π}{5 \cdot 4}$

Step 5.4.2

Multiply $5$ by $4$ .

Phase Shift: $- \frac{π}{20}$

Step 6

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{2 π}{5}$

Phase Shift: $- \frac{π}{20}$ ( $\frac{π}{20}$ to the left)

Vertical Shift: $- 3$

Step 7