Trigonometry Examples

$y = \cos (10 (x - π))$

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.

$a = 1$

$b = 10$

$c = 10 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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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: $1$

Period: $\frac{π}{5}$

Phase Shift: $π$ ( $π$ 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 $x = π$ .

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Step 6.1.1

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

$f (π) = \cos (10 (π) - 10 π)$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Subtract $10 π$ from $10 π$ .

$f (π) = \cos (0)$

Step 6.1.2.2

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

$f (π) = 1$

Step 6.1.2.3

The final answer is $1$ .

$1$

Step 6.2

Find the point at $x = \frac{21 π}{20}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{21 π}{20}$ in the expression.

$f (\frac{21 π}{20}) = \cos (10 (\frac{21 π}{20}) - 10 π)$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Cancel the common factor of $10$ .

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Step 6.2.2.1.1

Factor $10$ out of $20$ .

$f (\frac{21 π}{20}) = \cos (10 (\frac{21 π}{10 (2)}) - 10 π)$

Step 6.2.2.1.2

Cancel the common factor.

$f (\frac{21 π}{20}) = \cos (10 (\frac{21 π}{10 \cdot 2}) - 10 π)$

Step 6.2.2.1.3

Rewrite the expression.

$f (\frac{21 π}{20}) = \cos (\frac{21 π}{2} - 10 π)$

Step 6.2.2.2

To write $- 10 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{21 π}{20}) = \cos (\frac{21 π}{2} - 10 π \cdot \frac{2}{2})$

Step 6.2.2.3

Combine fractions.

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Step 6.2.2.3.1

Combine $- 10 π$ and $\frac{2}{2}$ .

$f (\frac{21 π}{20}) = \cos (\frac{21 π}{2} + \frac{- 10 π \cdot 2}{2})$

Step 6.2.2.3.2

Combine the numerators over the common denominator.

$f (\frac{21 π}{20}) = \cos (\frac{21 π - 10 π \cdot 2}{2})$

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Multiply $2$ by $- 10$ .

$f (\frac{21 π}{20}) = \cos (\frac{21 π - 20 π}{2})$

Step 6.2.2.4.2

Subtract $20 π$ from $21 π$ .

$f (\frac{21 π}{20}) = \cos (\frac{π}{2})$

Step 6.2.2.5

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{21 π}{20}) = 0$

Step 6.2.2.6

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{11 π}{10}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{11 π}{10}$ in the expression.

$f (\frac{11 π}{10}) = \cos (10 (\frac{11 π}{10}) - 10 π)$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Cancel the common factor of $10$ .

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Step 6.3.2.1.1

Cancel the common factor.

$f (\frac{11 π}{10}) = \cos (10 (\frac{11 π}{10}) - 10 π)$

Step 6.3.2.1.2

Rewrite the expression.

$f (\frac{11 π}{10}) = \cos (11 π - 10 π)$

Step 6.3.2.2

Subtract $10 π$ from $11 π$ .

$f (\frac{11 π}{10}) = \cos (π)$

Step 6.3.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (\frac{11 π}{10}) = - \cos (0)$

Step 6.3.2.4

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

$f (\frac{11 π}{10}) = - 1 \cdot 1$

Step 6.3.2.5

Multiply $- 1$ by $1$ .

$f (\frac{11 π}{10}) = - 1$

Step 6.3.2.6

The final answer is $- 1$ .

$- 1$

Step 6.4

Find the point at $x = \frac{23 π}{20}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{23 π}{20}$ in the expression.

$f (\frac{23 π}{20}) = \cos (10 (\frac{23 π}{20}) - 10 π)$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $10$ .

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Step 6.4.2.1.1

Factor $10$ out of $20$ .

$f (\frac{23 π}{20}) = \cos (10 (\frac{23 π}{10 (2)}) - 10 π)$

Step 6.4.2.1.2

Cancel the common factor.

$f (\frac{23 π}{20}) = \cos (10 (\frac{23 π}{10 \cdot 2}) - 10 π)$

Step 6.4.2.1.3

Rewrite the expression.

$f (\frac{23 π}{20}) = \cos (\frac{23 π}{2} - 10 π)$

Step 6.4.2.2

To write $- 10 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{23 π}{20}) = \cos (\frac{23 π}{2} - 10 π \cdot \frac{2}{2})$

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $- 10 π$ and $\frac{2}{2}$ .

$f (\frac{23 π}{20}) = \cos (\frac{23 π}{2} + \frac{- 10 π \cdot 2}{2})$

Step 6.4.2.3.2

Combine the numerators over the common denominator.

$f (\frac{23 π}{20}) = \cos (\frac{23 π - 10 π \cdot 2}{2})$

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply $2$ by $- 10$ .

$f (\frac{23 π}{20}) = \cos (\frac{23 π - 20 π}{2})$

Step 6.4.2.4.2

Subtract $20 π$ from $23 π$ .

$f (\frac{23 π}{20}) = \cos (\frac{3 π}{2})$

Step 6.4.2.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (\frac{23 π}{20}) = \cos (\frac{π}{2})$

Step 6.4.2.6

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{23 π}{20}) = 0$

Step 6.4.2.7

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{6 π}{5}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{6 π}{5}$ in the expression.

$f (\frac{6 π}{5}) = \cos (10 (\frac{6 π}{5}) - 10 π)$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Cancel the common factor of $5$ .

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Step 6.5.2.1.1.1

Factor $5$ out of $10$ .

$f (\frac{6 π}{5}) = \cos (5 (2) (\frac{6 π}{5}) - 10 π)$

Step 6.5.2.1.1.2

Cancel the common factor.

$f (\frac{6 π}{5}) = \cos (5 \cdot (2 (\frac{6 π}{5})) - 10 π)$

Step 6.5.2.1.1.3

Rewrite the expression.

$f (\frac{6 π}{5}) = \cos (2 (6 π) - 10 π)$

Step 6.5.2.1.2

Multiply $6$ by $2$ .

$f (\frac{6 π}{5}) = \cos (12 π - 10 π)$

Step 6.5.2.2

Subtract $10 π$ from $12 π$ .

$f (\frac{6 π}{5}) = \cos (2 π)$

Step 6.5.2.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{6 π}{5}) = \cos (0)$

Step 6.5.2.4

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

$f (\frac{6 π}{5}) = 1$

Step 6.5.2.5

The final answer is $1$ .

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ π & 1 \\ \frac{21 π}{20} & 0 \\ \frac{11 π}{10} & - 1 \\ \frac{23 π}{20} & 0 \\ \frac{6 π}{5} & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $\frac{π}{5}$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ π & 1 \\ \frac{21 π}{20} & 0 \\ \frac{11 π}{10} & - 1 \\ \frac{23 π}{20} & 0 \\ \frac{6 π}{5} & 1 \end{array}$

Step 8