Trigonometry Examples

$y = 2 \cos (0.5 x)$

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.

$a = 2$

$b = 0.5$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period of $2 \cos (0.5 x)$ .

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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 $0.5$ in the formula for period.

$\frac{2 π}{| 0.5 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.5$ is $0.5$ .

$\frac{2 π}{0.5}$

Step 3.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.5}$

Step 3.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.5}$

Step 3.6

Divide $6.2831853$ by $0.5$ .

$12.56637061$

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: $\frac{0}{0.5}$

Step 4.3

Divide $0$ by $0.5$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $12.56637061$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

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

$f (0) = 2 \cos (0.5 (0))$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $0.5$ by $0$ .

$f (0) = 2 \cos (0)$

Step 6.1.2.2

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

$f (0) = 2 \cdot 1$

Step 6.1.2.3

Multiply $2$ by $1$ .

$f (0) = 2$

Step 6.1.2.4

The final answer is $2$ .

$2$

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.

$f (3.14159265) = 2 \cos (0.5 (3.14159265))$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Multiply $0.5$ by $3.14159265$ .

$f (3.14159265) = 2 \cos (1.57079632)$

Step 6.2.2.2

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

$2 \cos (1.57079632)$

Step 6.2.3

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

$0$

Step 6.3

Find the point at $x = 6.2831853$ .

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Step 6.3.1

Replace the variable $x$ with $6.2831853$ in the expression.

$f (6.2831853) = 2 \cos (0.5 (6.2831853))$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Multiply $0.5$ by $6.2831853$ .

$f (6.2831853) = 2 \cos (3.14159265)$

Step 6.3.2.2

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

$2 \cos (3.14159265)$

Step 6.3.3

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

$- 2$

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.

$f (9.42477796) = 2 \cos (0.5 (9.42477796))$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Multiply $0.5$ by $9.42477796$ .

$f (9.42477796) = 2 \cos (4.71238898)$

Step 6.4.2.2

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

$2 \cos (4.71238898)$

Step 6.4.3

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

$0$

Step 6.5

Find the point at $x = 12.56637061$ .

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Step 6.5.1

Replace the variable $x$ with $12.56637061$ in the expression.

$f (12.56637061) = 2 \cos (0.5 (12.56637061))$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Multiply $0.5$ by $12.56637061$ .

$f (12.56637061) = 2 \cos (6.2831853)$

Step 6.5.2.2

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

$2 \cos (6.2831853)$

Step 6.5.3

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

$2$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 2 \\ 3.142 & 0 \\ 6.283 & - 2 \\ 9.425 & - 0 \\ 12.566 & 2 \end{array}$

Step 7

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

Amplitude: $2$

Period: $12.56637061$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 2 \\ 3.142 & 0 \\ 6.283 & - 2 \\ 9.425 & - 0 \\ 12.566 & 2 \end{array}$

Step 8