Trigonometry Examples

$y = 3 \cos (10 (x - π))$

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.

$a = 3$

$b = 10$

$c = 10 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $3 \cos (10 x - 10 π)$ .

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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 $10$ in the formula for period.

$\frac{2 π}{| 10 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$\frac{2 π}{10}$

Step 3.4

Cancel the common factor of $2$ and $10$ .

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Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{10}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $10$ .

$\frac{2 π}{2 \cdot 5}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 5}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{5}$

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: $\frac{10 π}{10}$

Step 4.3

Cancel the common factor of $10$ .

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Step 4.3.1

Cancel the common factor.

Phase Shift: $\frac{10 π}{10}$

Step 4.3.2

Divide $π$ by $1$ .

Phase Shift: $π$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{π}{5}$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: None

Step 6