Trigonometry Examples

$\cos (- 4 π x + 3) - 7$

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.

$a = 1$

$b = - 4 π$

$c = - 3$

$d = - 7$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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Step 3.1

Find the period of $\cos (- 4 π x + 3)$ .

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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 $- 4 π$ in the formula for period.

$\frac{2 π}{| - 4 π |}$

Step 3.1.3

$- 4 π$ is approximately $- 12.56637061$ which is negative so negate $- 4 π$ and remove the absolute value

$\frac{2 π}{4 π}$

Step 3.1.4

Cancel the common factor of $2$ and $4$ .

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Step 3.1.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4 π}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $4 π$ .

$\frac{2 (π)}{2 (2 π)}$

Step 3.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (2 π)}$

Step 3.1.4.2.3

Rewrite the expression.

$\frac{π}{2 π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Cancel the common factor.

$\frac{π}{2 π}$

Step 3.1.5.2

Rewrite the expression.

$\frac{1}{2}$

Step 3.2

Find the period of $- 7$ .

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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 $- 4 π$ in the formula for period.

$\frac{2 π}{| - 4 π |}$

Step 3.2.3

$- 4 π$ is approximately $- 12.56637061$ which is negative so negate $- 4 π$ and remove the absolute value

$\frac{2 π}{4 π}$

Step 3.2.4

Cancel the common factor of $2$ and $4$ .

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Step 3.2.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4 π}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $4 π$ .

$\frac{2 (π)}{2 (2 π)}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (2 π)}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{π}{2 π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Cancel the common factor.

$\frac{π}{2 π}$

Step 3.2.5.2

Rewrite the expression.

$\frac{1}{2}$

Step 3.3

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

$\frac{1}{2}$

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: $\frac{- 3}{- 4 π}$

Step 4.3

Dividing two negative values results in a positive value.

Phase Shift: $\frac{3}{4 π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{1}{2}$

Phase Shift: $\frac{3}{4 π}$ ( $\frac{3}{4 π}$ to the right)

Vertical Shift: $- 7$

Step 6