Trigonometry Examples

$4.4 \sin (0.16 π \cdot x - 0.6333) \cdot 3.6$

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.

$a = 15.84$

$b = 0.50265482$

$c = 0.6333$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $15.84$

Step 3

Find the period of $15.84 \sin (0.50265482 x - 0.6333)$ .

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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 $0.50265482$ in the formula for period.

$\frac{2 π}{| 0.50265482 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.50265482$ is $0.50265482$ .

$\frac{2 π}{0.50265482}$

Step 3.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.50265482}$

Step 3.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.50265482}$

Step 3.6

Divide $6.2831853$ by $0.50265482$ .

$12.5$

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: $\frac{0.6333}{0.50265482}$

Step 4.3

Divide $0.6333$ by $0.50265482$ .

Phase Shift: $1.25991031$

Step 5

List the properties of the trigonometric function.

Amplitude: $15.84$

Period: $12.5$

Phase Shift: $1.25991031$ ( $1.25991031$ to the right)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 1.25991031$ .

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Step 6.1.1

Replace the variable $x$ with $1.25991031$ in the expression.

$f (1.25991031) = 15.84 \sin (0.50265482 (1.25991031) - 0.6333)$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $0.50265482$ by $1.25991031$ .

$f (1.25991031) = 15.84 \sin (0.6333 - 0.6333)$

Step 6.1.2.2

Subtract $0.6333$ from $0.6333$ .

$f (1.25991031) = 15.84 \sin (0)$

Step 6.1.2.3

The exact value of $\sin (0)$ is $0$ .

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Step 6.1.2.3.1

Rewrite $0$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$f (1.25991031) = 15.84 \sin (\frac{0}{2})$

Step 6.1.2.3.2

Apply the sine half-angle identity.

$f (1.25991031) = 15.84 (\pm \sqrt{\frac{1 - \cos (0)}{2}})$

Step 6.1.2.3.3

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

$f (1.25991031) = 15.84 \sqrt{\frac{1 - \cos (0)}{2}}$

Step 6.1.2.3.4

Simplify $\sqrt{\frac{1 - \cos (0)}{2}}$ .

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Step 6.1.2.3.4.1

The exact value of $\cos (0)$ is $1$ .

$f (1.25991031) = 15.84 \sqrt{\frac{1 - 1 \cdot 1}{2}}$

Step 6.1.2.3.4.2

Multiply $- 1$ by $1$ .

$f (1.25991031) = 15.84 \sqrt{\frac{1 - 1}{2}}$

Step 6.1.2.3.4.3

Subtract $1$ from $1$ .

$f (1.25991031) = 15.84 \sqrt{\frac{0}{2}}$

Step 6.1.2.3.4.4

Divide $0$ by $2$ .

$f (1.25991031) = 15.84 \sqrt{0}$

Step 6.1.2.3.4.5

Rewrite $0$ as $0^{2}$ .

$f (1.25991031) = 15.84 \sqrt{0^{2}}$

Step 6.1.2.3.4.6

Pull terms out from under the radical, assuming positive real numbers.

$f (1.25991031) = 15.84 \cdot 0$

Step 6.1.2.4

Multiply $15.84$ by $0$ .

$f (1.25991031) = 0$

Step 6.1.2.5

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = 4.38491031$ .

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Step 6.2.1

Replace the variable $x$ with $4.38491031$ in the expression.

$f (4.38491031) = 15.84 \sin (0.50265482 (4.38491031) - 0.6333)$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify the expression.

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Step 6.2.2.1.1

Multiply $0.50265482$ by $4.38491031$ .

$f (4.38491031) = 15.84 \sin (2.20409632 - 0.6333)$

Step 6.2.2.1.2

Subtract $0.6333$ from $2.20409632$ .

$f (4.38491031) = 15.84 \sin (1.57079632)$

Step 6.2.2.2

Multiply $15.84$ by $\sin (1.57079632)$ .

$f (4.38491031) = 15.84$

Step 6.2.2.3

The final answer is $15.84$ .

$15.84$

Step 6.3

Find the point at $x = 7.50991031$ .

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Step 6.3.1

Replace the variable $x$ with $7.50991031$ in the expression.

$f (7.50991031) = 15.84 \sin (0.50265482 (7.50991031) - 0.6333)$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify the expression.

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Step 6.3.2.1.1

Multiply $0.50265482$ by $7.50991031$ .

$f (7.50991031) = 15.84 \sin (3.77489265 - 0.6333)$

Step 6.3.2.1.2

Subtract $0.6333$ from $3.77489265$ .

$f (7.50991031) = 15.84 \sin (3.14159265)$

Step 6.3.2.2

Multiply $15.84$ by $\sin (3.14159265)$ .

$f (7.50991031) = 0$

Step 6.3.2.3

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = 10.63491031$ .

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Step 6.4.1

Replace the variable $x$ with $10.63491031$ in the expression.

$f (10.63491031) = 15.84 \sin (0.50265482 (10.63491031) - 0.6333)$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify the expression.

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Step 6.4.2.1.1

Multiply $0.50265482$ by $10.63491031$ .

$f (10.63491031) = 15.84 \sin (5.34568898 - 0.6333)$

Step 6.4.2.1.2

Subtract $0.6333$ from $5.34568898$ .

$f (10.63491031) = 15.84 \sin (4.71238898)$

Step 6.4.2.2

Multiply $15.84$ by $\sin (4.71238898)$ .

$f (10.63491031) = - 15.84$

Step 6.4.2.3

The final answer is $- 15.84$ .

$- 15.84$

Step 6.5

Find the point at $x = 13.75991031$ .

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Step 6.5.1

Replace the variable $x$ with $13.75991031$ in the expression.

$f (13.75991031) = 15.84 \sin (0.50265482 (13.75991031) - 0.6333)$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify the expression.

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Step 6.5.2.1.1

Multiply $0.50265482$ by $13.75991031$ .

$f (13.75991031) = 15.84 \sin (6.9164853 - 0.6333)$

Step 6.5.2.1.2

Subtract $0.6333$ from $6.9164853$ .

$f (13.75991031) = 15.84 \sin (6.2831853)$

Step 6.5.2.2

Multiply $15.84$ by $\sin (6.2831853)$ .

$f (13.75991031) = 0$

Step 6.5.2.3

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 1.26 & 0 \\ 4.385 & 15.84 \\ 7.51 & 0 \\ 10.635 & - 15.84 \\ 13.76 & - 0 \end{array}$

Step 7

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

Amplitude: $15.84$

Period: $12.5$

Phase Shift: $1.25991031$ ( $1.25991031$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 1.26 & 0 \\ 4.385 & 15.84 \\ 7.51 & 0 \\ 10.635 & - 15.84 \\ 13.76 & - 0 \end{array}$

Step 8