Trigonometry Examples

$- 0.25 \cos (1.5 t - \frac{π}{3})$

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.

$a = - 0.25$

$b = 1.5$

$c = \frac{π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $0.25$

Step 3

Find the period of $- 0.25 \cos (1.5 x - \frac{π}{3})$ .

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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 $1.5$ in the formula for period.

$\frac{2 π}{| 1.5 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1.5$ is $1.5$ .

$\frac{2 π}{1.5}$

Step 3.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{1.5}$

Step 3.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{1.5}$

Step 3.6

Divide $6.2831853$ by $1.5$ .

$4.1887902$

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: $\frac{\frac{π}{3}}{1.5}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{3} \cdot \frac{1}{1.5}$

Step 4.4

Divide $1$ by $1.5$ .

Phase Shift: $\frac{π}{3} \cdot 0.\overline{6}$

Step 4.5

Multiply $\frac{π}{3} \cdot 0.\overline{6}$ .

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Step 4.5.1

Combine $\frac{π}{3}$ and $0.\overline{6}$ .

Phase Shift: $\frac{π \cdot 0.\overline{6}}{3}$

Step 4.5.2

Multiply $π$ by $0.\overline{6}$ .

Phase Shift: $\frac{2.0943951}{3}$

Step 4.6

Divide $2.0943951$ by $3$ .

Phase Shift: $0.6981317$

Step 5

List the properties of the trigonometric function.

Amplitude: $0.25$

Period: $4.1887902$

Phase Shift: $0.6981317$ ( $0.6981317$ 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 $x = 0.6981317$ .

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Step 6.1.1

Replace the variable $x$ with $0.6981317$ in the expression.

$f (0.6981317) = - 0.25 \cos (1.5 (0.6981317) - \frac{π}{3})$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $1.5$ by $0.6981317$ .

$f (0.6981317) = - 0.25 \cos (1.04719755 - \frac{π}{3})$

Step 6.1.2.2

Subtract $\frac{π}{3}$ from $1.04719755$ .

$f (0.6981317) = - 0.25 \cos (0)$

Step 6.1.2.3

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

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Step 6.1.2.3.1

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

$f (0.6981317) = - 0.25 \cos (\frac{0}{2})$

Step 6.1.2.3.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$f (0.6981317) = - 0.25 (\pm \sqrt{\frac{1 + \cos (0)}{2}})$

Step 6.1.2.3.3

Change the $\pm$ to $+$ because cosine is positive in the fourth quadrant.

$f (0.6981317) = - 0.25 \sqrt{\frac{1 + \cos (0)}{2}}$

Step 6.1.2.3.4

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

$f (0.6981317) = - 0.25 \sqrt{\frac{1 + 1}{2}}$

Step 6.1.2.3.5

Simplify $\sqrt{\frac{1 + 1}{2}}$ .

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Step 6.1.2.3.5.1

Add $1$ and $1$ .

$f (0.6981317) = - 0.25 \sqrt{\frac{2}{2}}$

Step 6.1.2.3.5.2

Divide $2$ by $2$ .

$f (0.6981317) = - 0.25 \sqrt{1}$

Step 6.1.2.3.5.3

Any root of $1$ is $1$ .

$f (0.6981317) = - 0.25 \cdot 1$

Step 6.1.2.4

Multiply $- 0.25$ by $1$ .

$f (0.6981317) = - 0.25$

Step 6.1.2.5

The final answer is $- 0.25$ .

$- 0.25$

Step 6.2

Find the point at $x = 1.74532925$ .

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Step 6.2.1

Replace the variable $x$ with $1.74532925$ in the expression.

$f (1.74532925) = - 0.25 \cos (1.5 (1.74532925) - \frac{π}{3})$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Multiply $1.5$ by $1.74532925$ .

$f (1.74532925) = - 0.25 \cos (2.61799387 - \frac{π}{3})$

Step 6.2.2.2

Subtract $\frac{π}{3}$ from $2.61799387$ .

$f (1.74532925) = - 0.25 \cos (1.57079632)$

Step 6.2.2.3

Multiply $- 0.25$ by $\cos (1.57079632)$ .

$f (1.74532925) = 0$

Step 6.2.2.4

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = 2.7925268$ .

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Step 6.3.1

Replace the variable $x$ with $2.7925268$ in the expression.

$f (2.7925268) = - 0.25 \cos (1.5 (2.7925268) - \frac{π}{3})$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Multiply $1.5$ by $2.7925268$ .

$f (2.7925268) = - 0.25 \cos (4.1887902 - \frac{π}{3})$

Step 6.3.2.2

Subtract $\frac{π}{3}$ from $4.1887902$ .

$f (2.7925268) = - 0.25 \cos (3.14159265)$

Step 6.3.2.3

Multiply $- 0.25$ by $\cos (3.14159265)$ .

$f (2.7925268) = 0.25$

Step 6.3.2.4

The final answer is $0.25$ .

$0.25$

Step 6.4

Find the point at $x = 3.83972435$ .

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Step 6.4.1

Replace the variable $x$ with $3.83972435$ in the expression.

$f (3.83972435) = - 0.25 \cos (1.5 (3.83972435) - \frac{π}{3})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Multiply $1.5$ by $3.83972435$ .

$f (3.83972435) = - 0.25 \cos (5.75958653 - \frac{π}{3})$

Step 6.4.2.2

Subtract $\frac{π}{3}$ from $5.75958653$ .

$f (3.83972435) = - 0.25 \cos (4.71238898)$

Step 6.4.2.3

Multiply $- 0.25$ by $\cos (4.71238898)$ .

$f (3.83972435) = 0$

Step 6.4.2.4

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = 4.8869219$ .

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Step 6.5.1

Replace the variable $x$ with $4.8869219$ in the expression.

$f (4.8869219) = - 0.25 \cos (1.5 (4.8869219) - \frac{π}{3})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Multiply $1.5$ by $4.8869219$ .

$f (4.8869219) = - 0.25 \cos (7.33038285 - \frac{π}{3})$

Step 6.5.2.2

Subtract $\frac{π}{3}$ from $7.33038285$ .

$f (4.8869219) = - 0.25 \cos (6.2831853)$

Step 6.5.2.3

Multiply $- 0.25$ by $\cos (6.2831853)$ .

$f (4.8869219) = - 0.25$

Step 6.5.2.4

The final answer is $- 0.25$ .

$- 0.25$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0.698 & - 0.25 \\ 1.745 & - 0 \\ 2.793 & 0.25 \\ 3.84 & 0 \\ 4.887 & - 0.25 \end{array}$

Step 7

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

Amplitude: $0.25$

Period: $4.1887902$

Phase Shift: $0.6981317$ ( $0.6981317$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0.698 & - 0.25 \\ 1.745 & - 0 \\ 2.793 & 0.25 \\ 3.84 & 0 \\ 4.887 & - 0.25 \end{array}$

Step 8