Trigonometry Examples

$h (x) = 0.5 \sin (0.25 x + 0.75 π) - 5$

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.

$a = 0.5$

$b = 0.25$

$c = - 2.35619449$

$d = - 5$

Step 2

Find the amplitude $| a |$ .

Amplitude: $0.5$

Step 3

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

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Step 3.1

Find the period of $0.5 \sin (0.25 x + 2.35619449)$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $0.25$ in the formula for period.

$\frac{2 π}{| 0.25 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.25$ is $0.25$ .

$\frac{2 π}{0.25}$

Step 3.1.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.25}$

Step 3.1.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.25}$

Step 3.1.6

Divide $6.2831853$ by $0.25$ .

$25.13274122$

Step 3.2

Find the period of $- 5$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $0.25$ in the formula for period.

$\frac{2 π}{| 0.25 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.25$ is $0.25$ .

$\frac{2 π}{0.25}$

Step 3.2.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.25}$

Step 3.2.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.25}$

Step 3.2.6

Divide $6.2831853$ by $0.25$ .

$25.13274122$

Step 3.3

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

$25.13274122$

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: $\frac{- 2.35619449}{0.25}$

Step 4.3

Divide $- 2.35619449$ by $0.25$ .

Phase Shift: $- 9.42477796$

Step 5

List the properties of the trigonometric function.

Amplitude: $0.5$

Period: $25.13274122$

Phase Shift: $- 9.42477796$ ( $9.42477796$ to the left)

Vertical Shift: $- 5$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - 9.42477796$ .

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Step 6.1.1

Replace the variable $x$ with $- 9.42477796$ in the expression.

$f (- 9.42477796) = 0.5 \sin (0.25 (- 9.42477796) + 2.35619449) - 5$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Multiply $0.25$ by $- 9.42477796$ .

$f (- 9.42477796) = 0.5 \sin (- 2.35619449 + 2.35619449) - 5$

Step 6.1.2.1.2

Add $- 2.35619449$ and $2.35619449$ .

$f (- 9.42477796) = 0.5 \sin (0) - 5$

Step 6.1.2.1.3

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

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Step 6.1.2.1.3.1

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

$f (- 9.42477796) = 0.5 \sin (\frac{0}{2}) - 5$

Step 6.1.2.1.3.2

Apply the sine half-angle identity.

$f (- 9.42477796) = 0.5 (\pm \sqrt{\frac{1 - \cos (0)}{2}}) - 5$

Step 6.1.2.1.3.3

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

$f (- 9.42477796) = 0.5 \sqrt{\frac{1 - \cos (0)}{2}} - 5$

Step 6.1.2.1.3.4

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

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Step 6.1.2.1.3.4.1

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

$f (- 9.42477796) = 0.5 \sqrt{\frac{1 - 1 \cdot 1}{2}} - 5$

Step 6.1.2.1.3.4.2

Multiply $- 1$ by $1$ .

$f (- 9.42477796) = 0.5 \sqrt{\frac{1 - 1}{2}} - 5$

Step 6.1.2.1.3.4.3

Subtract $1$ from $1$ .

$f (- 9.42477796) = 0.5 \sqrt{\frac{0}{2}} - 5$

Step 6.1.2.1.3.4.4

Divide $0$ by $2$ .

$f (- 9.42477796) = 0.5 \sqrt{0} - 5$

Step 6.1.2.1.3.4.5

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

$f (- 9.42477796) = 0.5 \sqrt{0^{2}} - 5$

Step 6.1.2.1.3.4.6

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

$f (- 9.42477796) = 0.5 \cdot 0 - 5$

Step 6.1.2.1.4

Multiply $0.5$ by $0$ .

$f (- 9.42477796) = 0 - 5$

Step 6.1.2.2

Subtract $5$ from $0$ .

$f (- 9.42477796) = - 5$

Step 6.1.2.3

The final answer is $- 5$ .

$- 5$

Step 6.2

Find the point at $x = - 3.14159265$ .

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Step 6.2.1

Replace the variable $x$ with $- 3.14159265$ in the expression.

$f (- 3.14159265) = 0.5 \sin (0.25 (- 3.14159265) + 2.35619449) - 5$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $0.25$ by $- 3.14159265$ .

$f (- 3.14159265) = 0.5 \sin (- 0.78539816 + 2.35619449) - 5$

Step 6.2.2.1.2

Add $- 0.78539816$ and $2.35619449$ .

$f (- 3.14159265) = 0.5 \sin (1.57079632) - 5$

Step 6.2.2.1.3

Multiply $0.5$ by $\sin (1.57079632)$ .

$f (- 3.14159265) = 0.5 - 5$

Step 6.2.2.2

Subtract $5$ from $0.5$ .

$f (- 3.14159265) = - 4.5$

Step 6.2.2.3

The final answer is $- 4.5$ .

$- 4.5$

Step 6.3

Find the point at $x = 3.14159265$ .

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Step 6.3.1

Replace the variable $x$ with $3.14159265$ in the expression.

$f (3.14159265) = 0.5 \sin (0.25 (3.14159265) + 2.35619449) - 5$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Multiply $0.25$ by $3.14159265$ .

$f (3.14159265) = 0.5 \sin (0.78539816 + 2.35619449) - 5$

Step 6.3.2.1.2

Add $0.78539816$ and $2.35619449$ .

$f (3.14159265) = 0.5 \sin (3.14159265) - 5$

Step 6.3.2.1.3

Multiply $0.5$ by $\sin (3.14159265)$ .

$f (3.14159265) = 0 - 5$

Step 6.3.2.2

Subtract $5$ from $0$ .

$f (3.14159265) = - 5$

Step 6.3.2.3

The final answer is $- 5$ .

$- 5$

Step 6.4

Find the point at $x = 9.42477796$ .

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Step 6.4.1

Replace the variable $x$ with $9.42477796$ in the expression.

$f (9.42477796) = 0.5 \sin (0.25 (9.42477796) + 2.35619449) - 5$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Multiply $0.25$ by $9.42477796$ .

$f (9.42477796) = 0.5 \sin (2.35619449 + 2.35619449) - 5$

Step 6.4.2.1.2

Add $2.35619449$ and $2.35619449$ .

$f (9.42477796) = 0.5 \sin (4.71238898) - 5$

Step 6.4.2.1.3

Multiply $0.5$ by $\sin (4.71238898)$ .

$f (9.42477796) = - 0.5 - 5$

Step 6.4.2.2

Subtract $5$ from $- 0.5$ .

$f (9.42477796) = - 5.5$

Step 6.4.2.3

The final answer is $- 5.5$ .

$- 5.5$

Step 6.5

Find the point at $x = 15.70796326$ .

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Step 6.5.1

Replace the variable $x$ with $15.70796326$ in the expression.

$f (15.70796326) = 0.5 \sin (0.25 (15.70796326) + 2.35619449) - 5$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Multiply $0.25$ by $15.70796326$ .

$f (15.70796326) = 0.5 \sin (3.92699081 + 2.35619449) - 5$

Step 6.5.2.1.2

Add $3.92699081$ and $2.35619449$ .

$f (15.70796326) = 0.5 \sin (6.2831853) - 5$

Step 6.5.2.1.3

Multiply $0.5$ by $\sin (6.2831853)$ .

$f (15.70796326) = 0 - 5$

Step 6.5.2.2

Subtract $5$ from $0$ .

$f (15.70796326) = - 5$

Step 6.5.2.3

The final answer is $- 5$ .

$- 5$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 9.425 & - 5 \\ - 3.142 & - 4.5 \\ 3.142 & - 5 \\ 9.425 & - 5.5 \\ 15.708 & - 5 \end{array}$

Step 7

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

Amplitude: $0.5$

Period: $25.13274122$

Phase Shift: $- 9.42477796$ ( $9.42477796$ to the left)

Vertical Shift: $- 5$

$\begin{array}{c} x & f (x) \\ - 9.425 & - 5 \\ - 3.142 & - 4.5 \\ 3.142 & - 5 \\ 9.425 & - 5.5 \\ 15.708 & - 5 \end{array}$

Step 8