Trigonometry Examples

$1.1 \sin (x - 1.9)$

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.

$a = 1.1$

$b = 1$

$c = 1.9$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1.1$

Step 3

Find the period of $1.1 \sin (x - 1.9)$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.3

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

$\frac{2 π}{1}$

Step 3.4

Divide $2 π$ by $1$ .

$2 π$

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: $\frac{1.9}{1}$

Step 4.3

Divide $1.9$ by $1$ .

Phase Shift: $1.9$

Step 5

List the properties of the trigonometric function.

Amplitude: $1.1$

Period: $2 π$

Phase Shift: $1.9$ ( $1.9$ 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 $x = 1.9$ .

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Step 6.1.1

Replace the variable $x$ with $1.9$ in the expression.

$f (1.9) = 1.1 \sin ((1.9) - 1.9)$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Subtract $1.9$ from $1.9$ .

$f (1.9) = 1.1 \sin (0)$

Step 6.1.2.2

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

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Step 6.1.2.2.1

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

$f (1.9) = 1.1 \sin (\frac{0}{2})$

Step 6.1.2.2.2

Apply the sine half-angle identity.

$f (1.9) = 1.1 (\pm \sqrt{\frac{1 - \cos (0)}{2}})$

Step 6.1.2.2.3

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

$f (1.9) = 1.1 \sqrt{\frac{1 - \cos (0)}{2}}$

Step 6.1.2.2.4

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

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Step 6.1.2.2.4.1

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

$f (1.9) = 1.1 \sqrt{\frac{1 - 1 \cdot 1}{2}}$

Step 6.1.2.2.4.2

Multiply $- 1$ by $1$ .

$f (1.9) = 1.1 \sqrt{\frac{1 - 1}{2}}$

Step 6.1.2.2.4.3

Subtract $1$ from $1$ .

$f (1.9) = 1.1 \sqrt{\frac{0}{2}}$

Step 6.1.2.2.4.4

Divide $0$ by $2$ .

$f (1.9) = 1.1 \sqrt{0}$

Step 6.1.2.2.4.5

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

$f (1.9) = 1.1 \sqrt{0^{2}}$

Step 6.1.2.2.4.6

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

$f (1.9) = 1.1 \cdot 0$

Step 6.1.2.3

Multiply $1.1$ by $0$ .

$f (1.9) = 0$

Step 6.1.2.4

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = 3.47079632$ .

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Step 6.2.1

Replace the variable $x$ with $3.47079632$ in the expression.

$f (3.47079632) = 1.1 \sin ((3.47079632) - 1.9)$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Subtract $1.9$ from $3.47079632$ .

$f (3.47079632) = 1.1 \sin (1.57079632)$

Step 6.2.2.2

Multiply $1.1$ by $\sin (1.57079632)$ .

$f (3.47079632) = 1.1$

Step 6.2.2.3

The final answer is $1.1$ .

$1.1$

Step 6.3

Find the point at $x = 5.04159265$ .

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Step 6.3.1

Replace the variable $x$ with $5.04159265$ in the expression.

$f (5.04159265) = 1.1 \sin ((5.04159265) - 1.9)$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Subtract $1.9$ from $5.04159265$ .

$f (5.04159265) = 1.1 \sin (3.14159265)$

Step 6.3.2.2

Multiply $1.1$ by $\sin (3.14159265)$ .

$f (5.04159265) = 0$

Step 6.3.2.3

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = 6.61238898$ .

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Step 6.4.1

Replace the variable $x$ with $6.61238898$ in the expression.

$f (6.61238898) = 1.1 \sin ((6.61238898) - 1.9)$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Subtract $1.9$ from $6.61238898$ .

$f (6.61238898) = 1.1 \sin (4.71238898)$

Step 6.4.2.2

Multiply $1.1$ by $\sin (4.71238898)$ .

$f (6.61238898) = - 1.1$

Step 6.4.2.3

The final answer is $- 1.1$ .

$- 1.1$

Step 6.5

Find the point at $x = 8.1831853$ .

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Step 6.5.1

Replace the variable $x$ with $8.1831853$ in the expression.

$f (8.1831853) = 1.1 \sin ((8.1831853) - 1.9)$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Subtract $1.9$ from $8.1831853$ .

$f (8.1831853) = 1.1 \sin (6.2831853)$

Step 6.5.2.2

Multiply $1.1$ by $\sin (6.2831853)$ .

$f (8.1831853) = 0$

Step 6.5.2.3

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 1.9 & 0 \\ 3.471 & 1.1 \\ 5.042 & 0 \\ 6.612 & - 1.1 \\ 8.183 & 0 \end{array}$

Step 7

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

Amplitude: $1.1$

Period: $2 π$

Phase Shift: $1.9$ ( $1.9$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 1.9 & 0 \\ 3.471 & 1.1 \\ 5.042 & 0 \\ 6.612 & - 1.1 \\ 8.183 & 0 \end{array}$

Step 8