Trigonometry Examples

$g (x) = 2 \sin (x + 0.08)$

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.

$a = 2$

$b = 1$

$c = - 0.08$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period of $2 \sin (x + 0.08)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Step 4

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

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

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

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

Phase Shift: $\frac{- 0.08}{1}$

Divide $- 0.08$ by $1$ .

Phase Shift: $- 0.08$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $2 π$

Phase Shift: $- 0.08$ ( $0.08$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

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Find the point at $x = - 0.08$ .

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Replace the variable $x$ with $- 0.08$ in the expression.

$f (- 0.08) = 2 \sin ((- 0.08) + 0.08)$

Simplify the result.

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Add $- 0.08$ and $0.08$ .

$f (- 0.08) = 2 \sin (0)$

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

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

$f (- 0.08) = 2 \sin (\frac{0}{2})$

Apply the sine half-angle identity.

$f (- 0.08) = 2 (\pm \sqrt{\frac{1 - \cos (0)}{2}})$

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

$f (- 0.08) = 2 \sqrt{\frac{1 - \cos (0)}{2}}$

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

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

$f (- 0.08) = 2 \sqrt{\frac{1 - 1 \cdot 1}{2}}$

Multiply $- 1$ by $1$ .

$f (- 0.08) = 2 \sqrt{\frac{1 - 1}{2}}$

Subtract $1$ from $1$ .

$f (- 0.08) = 2 \sqrt{\frac{0}{2}}$

Divide $0$ by $2$ .

$f (- 0.08) = 2 \sqrt{0}$

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

$f (- 0.08) = 2 \sqrt{0^{2}}$

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

$f (- 0.08) = 2 \cdot 0$

Multiply $2$ by $0$ .

$f (- 0.08) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 1.49079632$ .

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Replace the variable $x$ with $1.49079632$ in the expression.

$f (1.49079632) = 2 \sin ((1.49079632) + 0.08)$

Simplify the result.

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Add $1.49079632$ and $0.08$ .

$f (1.49079632) = 2 \sin (1.57079632)$

The final answer is $2 \sin (1.57079632)$ .

$2 \sin (1.57079632)$

Convert $2 \sin (1.57079632)$ to a decimal.

$2$

Find the point at $x = 3.06159265$ .

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Replace the variable $x$ with $3.06159265$ in the expression.

$f (3.06159265) = 2 \sin ((3.06159265) + 0.08)$

Simplify the result.

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Add $3.06159265$ and $0.08$ .

$f (3.06159265) = 2 \sin (3.14159265)$

The final answer is $2 \sin (3.14159265)$ .

$2 \sin (3.14159265)$

Convert $2 \sin (3.14159265)$ to a decimal.

$0$

Find the point at $x = 4.63238898$ .

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Replace the variable $x$ with $4.63238898$ in the expression.

$f (4.63238898) = 2 \sin ((4.63238898) + 0.08)$

Simplify the result.

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Add $4.63238898$ and $0.08$ .

$f (4.63238898) = 2 \sin (4.71238898)$

The final answer is $2 \sin (4.71238898)$ .

$2 \sin (4.71238898)$

Convert $2 \sin (4.71238898)$ to a decimal.

$- 2$

Find the point at $x = 6.2031853$ .

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Replace the variable $x$ with $6.2031853$ in the expression.

$f (6.2031853) = 2 \sin ((6.2031853) + 0.08)$

Simplify the result.

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Add $6.2031853$ and $0.08$ .

$f (6.2031853) = 2 \sin (6.2831853)$

The final answer is $2 \sin (6.2831853)$ .

$2 \sin (6.2831853)$

Convert $2 \sin (6.2831853)$ to a decimal.

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 0.08 & 0 \\ 1.491 & 2 \\ 3.062 & 0 \\ 4.632 & - 2 \\ 6.203 & 0 \end{array}$

Step 7

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

Amplitude: $2$

Period: $2 π$

Phase Shift: $- 0.08$ ( $0.08$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 0.08 & 0 \\ 1.491 & 2 \\ 3.062 & 0 \\ 4.632 & - 2 \\ 6.203 & 0 \end{array}$

Step 8