Trigonometry Examples

$\cos (- 0.4 x)$

Step 1

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

$a = 1$

$b = - 0.4$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $\cos (- 0.4 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $- 0.4$ in the formula for period.

$\frac{2 π}{| - 0.4 |}$

The absolute value is the distance between a number and zero. The distance between $- 0.4$ and $0$ is $0.4$ .

$\frac{2 π}{0.4}$

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.4}$

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.4}$

Divide $6.2831853$ by $0.4$ .

$15.70796326$

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: $\frac{0}{- 0.4}$

Divide $0$ by $- 0.4$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $15.70796326$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Find the point at $x = 0$ .

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Replace the variable $x$ with $0$ in the expression.

$f (0) = \cos (- 0.4 \cdot 0)$

Simplify the result.

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Multiply $- 0.4$ by $0$ .

$f (0) = \cos (0)$

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

$f (0) = 1$

The final answer is $1$ .

$1$

Find the point at $x = 3.92699081$ .

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Replace the variable $x$ with $3.92699081$ in the expression.

$f (3.92699081) = \cos (- 0.4 \cdot 3.92699081)$

Simplify the result.

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Multiply $- 0.4$ by $3.92699081$ .

$f (3.92699081) = \cos (- 1.57079632)$

The final answer is $\cos (- 1.57079632)$ .

$\cos (- 1.57079632)$

Convert $\cos (- 1.57079632)$ to a decimal.

$0$

Find the point at $x = 7.85398163$ .

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Replace the variable $x$ with $7.85398163$ in the expression.

$f (7.85398163) = \cos (- 0.4 \cdot 7.85398163)$

Simplify the result.

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Multiply $- 0.4$ by $7.85398163$ .

$f (7.85398163) = \cos (- 3.14159265)$

The final answer is $\cos (- 3.14159265)$ .

$\cos (- 3.14159265)$

Convert $\cos (- 3.14159265)$ to a decimal.

$- 1$

Find the point at $x = 11.78097245$ .

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Replace the variable $x$ with $11.78097245$ in the expression.

$f (11.78097245) = \cos (- 0.4 \cdot 11.78097245)$

Simplify the result.

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Multiply $- 0.4$ by $11.78097245$ .

$f (11.78097245) = \cos (- 4.71238898)$

The final answer is $\cos (- 4.71238898)$ .

$\cos (- 4.71238898)$

Convert $\cos (- 4.71238898)$ to a decimal.

$0$

Find the point at $x = 15.70796326$ .

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

$f (15.70796326) = \cos (- 0.4 \cdot 15.70796326)$

Simplify the result.

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Multiply $- 0.4$ by $15.70796326$ .

$f (15.70796326) = \cos (- 6.2831853)$

The final answer is $\cos (- 6.2831853)$ .

$\cos (- 6.2831853)$

Convert $\cos (- 6.2831853)$ to a decimal.

$1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ 3.927 & 0 \\ 7.854 & - 1 \\ 11.781 & - 0 \\ 15.708 & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $15.70796326$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ 3.927 & 0 \\ 7.854 & - 1 \\ 11.781 & - 0 \\ 15.708 & 1 \end{array}$

Step 8