Trigonometry Examples

$y = - 1 + 6 \cos (\frac{2 π}{7} \cdot (x - 5))$

Step 1

Rewrite the expression as $6 \cos (\frac{2 π x}{7} - \frac{10 π}{7}) - 1$ .

$6 \cos (\frac{2 π x}{7} - \frac{10 π}{7}) - 1$

Step 2

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 6$

$b = \frac{2 π}{7}$

$c = \frac{10 π}{7}$

$d = - 1$

Step 3

Find the amplitude $| a |$ .

Amplitude: $6$

Step 4

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

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Step 4.1

Find the period of $6 \cos (\frac{2 π x}{7} - \frac{10 π}{7})$ .

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Step 4.1.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.1.2

Replace $b$ with $\frac{2 π}{7}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{7} |}$

Step 4.1.3

$\frac{2 π}{7}$ is approximately $0.8975979$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{7}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{2 π}$

Step 4.1.5

Cancel the common factor of $2 π$ .

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Step 4.1.5.1

Cancel the common factor.

$2 π \frac{7}{2 π}$

Step 4.1.5.2

Rewrite the expression.

$7$

Step 4.2

Find the period of $- 1$ .

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Step 4.2.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.2

Replace $b$ with $\frac{2 π}{7}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{7} |}$

Step 4.2.3

$\frac{2 π}{7}$ is approximately $0.8975979$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{7}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{2 π}$

Step 4.2.5

Cancel the common factor of $2 π$ .

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Step 4.2.5.1

Cancel the common factor.

$2 π \frac{7}{2 π}$

Step 4.2.5.2

Rewrite the expression.

$7$

Step 4.3

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

$7$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{\frac{10 π}{7}}{\frac{2 π}{7}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{10 π}{7} \cdot \frac{7}{2 π}$

Step 5.4

Cancel the common factor of $2 π$ .

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Step 5.4.1

Factor $2 π$ out of $10 π$ .

Phase Shift: $\frac{2 π (5)}{7} \cdot \frac{7}{2 π}$

Step 5.4.2

Cancel the common factor.

Phase Shift: $\frac{2 π \cdot 5}{7} \cdot \frac{7}{2 π}$

Step 5.4.3

Rewrite the expression.

Phase Shift: $\frac{5}{7} \cdot 7$

Step 5.5

Cancel the common factor of $7$ .

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Step 5.5.1

Cancel the common factor.

Phase Shift: $\frac{5}{7} \cdot 7$

Step 5.5.2

Rewrite the expression.

Phase Shift: $5$

Step 6

List the properties of the trigonometric function.

Amplitude: $6$

Period: $7$

Phase Shift: $5$ ( $5$ to the right)

Vertical Shift: $- 1$

Step 7

Select a few points to graph.

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Step 7.1

Find the point at $x = 5$ .

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Step 7.1.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - 1 + 6 \cos (\frac{2 π (5)}{7} - \frac{10 π}{7})$

Step 7.1.2

Simplify the result.

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Step 7.1.2.1

Simplify each term.

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Step 7.1.2.1.1

Combine the numerators over the common denominator.

$f (5) = - 1 + 6 \cos (\frac{2 π (5) - 10 π}{7})$

Step 7.1.2.1.2

Multiply $5$ by $2$ .

$f (5) = - 1 + 6 \cos (\frac{10 π - 10 π}{7})$

Step 7.1.2.1.3

Subtract $10 π$ from $10 π$ .

$f (5) = - 1 + 6 \cos (\frac{0}{7})$

Step 7.1.2.1.4

Divide $0$ by $7$ .

$f (5) = - 1 + 6 \cos (0)$

Step 7.1.2.1.5

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

$f (5) = - 1 + 6 \cdot 1$

Step 7.1.2.1.6

Multiply $6$ by $1$ .

$f (5) = - 1 + 6$

Step 7.1.2.2

Add $- 1$ and $6$ .

$f (5) = 5$

Step 7.1.2.3

The final answer is $5$ .

$5$

Step 7.2

Find the point at $x = \frac{27}{4}$ .

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Step 7.2.1

Replace the variable $x$ with $\frac{27}{4}$ in the expression.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{2 π (\frac{27}{4})}{7} - \frac{10 π}{7})$

Step 7.2.2

Simplify the result.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Combine the numerators over the common denominator.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{2 π (\frac{27}{4}) - 10 π}{7})$

Step 7.2.2.1.2

Simplify each term.

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Step 7.2.2.1.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.2.1.1

Factor $2$ out of $2 π$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{2 (π) (\frac{27}{4}) - 10 π}{7})$

Step 7.2.2.1.2.1.2

Factor $2$ out of $4$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{2 (π) (\frac{27}{2 (2)}) - 10 π}{7})$

Step 7.2.2.1.2.1.3

Cancel the common factor.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{2 π (\frac{27}{2 \cdot 2}) - 10 π}{7})$

Step 7.2.2.1.2.1.4

Rewrite the expression.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{π (\frac{27}{2}) - 10 π}{7})$

Step 7.2.2.1.2.2

Combine $π$ and $\frac{27}{2}$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{π \cdot 27}{2} - 10 π}{7})$

Step 7.2.2.1.2.3

Move $27$ to the left of $π$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{27 π}{2} - 10 π}{7})$

Step 7.2.2.1.3

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

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{27 π}{2} - 10 π \cdot \frac{2}{2}}{7})$

Step 7.2.2.1.4

Combine $- 10 π$ and $\frac{2}{2}$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{27 π}{2} + \frac{- 10 π \cdot 2}{2}}{7})$

Step 7.2.2.1.5

Combine the numerators over the common denominator.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{27 π - 10 π \cdot 2}{2}}{7})$

Step 7.2.2.1.6

Simplify the numerator.

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Step 7.2.2.1.6.1

Multiply $2$ by $- 10$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{27 π - 20 π}{2}}{7})$

Step 7.2.2.1.6.2

Subtract $20 π$ from $27 π$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{\frac{7 π}{2}}{7})$

Step 7.2.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{7 π}{2} \cdot \frac{1}{7})$

Step 7.2.2.1.8

Cancel the common factor of $7$ .

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Step 7.2.2.1.8.1

Factor $7$ out of $7 π$ .

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{7 (π)}{2} \cdot \frac{1}{7})$

Step 7.2.2.1.8.2

Cancel the common factor.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{7 π}{2} \cdot \frac{1}{7})$

Step 7.2.2.1.8.3

Rewrite the expression.

$f (\frac{27}{4}) = - 1 + 6 \cos (\frac{π}{2})$

Step 7.2.2.1.9

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

$f (\frac{27}{4}) = - 1 + 6 \cdot 0$

Step 7.2.2.1.10

Multiply $6$ by $0$ .

$f (\frac{27}{4}) = - 1 + 0$

Step 7.2.2.2

Add $- 1$ and $0$ .

$f (\frac{27}{4}) = - 1$

Step 7.2.2.3

The final answer is $- 1$ .

$- 1$

Step 7.3

Find the point at $x = \frac{17}{2}$ .

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Step 7.3.1

Replace the variable $x$ with $\frac{17}{2}$ in the expression.

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{2 π (\frac{17}{2})}{7} - \frac{10 π}{7})$

Step 7.3.2

Simplify the result.

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Step 7.3.2.1

Simplify each term.

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Step 7.3.2.1.1

Combine the numerators over the common denominator.

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{2 π (\frac{17}{2}) - 10 π}{7})$

Step 7.3.2.1.2

Simplify each term.

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Step 7.3.2.1.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.2.1.1

Factor $2$ out of $2 π$ .

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{2 (π) (\frac{17}{2}) - 10 π}{7})$

Step 7.3.2.1.2.1.2

Cancel the common factor.

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{2 π (\frac{17}{2}) - 10 π}{7})$

Step 7.3.2.1.2.1.3

Rewrite the expression.

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{π \cdot 17 - 10 π}{7})$

Step 7.3.2.1.2.2

Move $17$ to the left of $π$ .

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{17 π - 10 π}{7})$

Step 7.3.2.1.3

Subtract $10 π$ from $17 π$ .

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{7 π}{7})$

Step 7.3.2.1.4

Cancel the common factor of $7$ .

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Step 7.3.2.1.4.1

Cancel the common factor.

$f (\frac{17}{2}) = - 1 + 6 \cos (\frac{7 π}{7})$

Step 7.3.2.1.4.2

Divide $π$ by $1$ .

$f (\frac{17}{2}) = - 1 + 6 \cos (π)$

Step 7.3.2.1.5

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.

$f (\frac{17}{2}) = - 1 + 6 (- \cos (0))$

Step 7.3.2.1.6

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

$f (\frac{17}{2}) = - 1 + 6 (- 1 \cdot 1)$

Step 7.3.2.1.7

Multiply $6 (- 1 \cdot 1)$ .

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Step 7.3.2.1.7.1

Multiply $- 1$ by $1$ .

$f (\frac{17}{2}) = - 1 + 6 \cdot - 1$

Step 7.3.2.1.7.2

Multiply $6$ by $- 1$ .

$f (\frac{17}{2}) = - 1 - 6$

Step 7.3.2.2

Subtract $6$ from $- 1$ .

$f (\frac{17}{2}) = - 7$

Step 7.3.2.3

The final answer is $- 7$ .

$- 7$

Step 7.4

Find the point at $x = \frac{41}{4}$ .

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Step 7.4.1

Replace the variable $x$ with $\frac{41}{4}$ in the expression.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{2 π (\frac{41}{4})}{7} - \frac{10 π}{7})$

Step 7.4.2

Simplify the result.

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Step 7.4.2.1

Simplify each term.

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Step 7.4.2.1.1

Combine the numerators over the common denominator.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{2 π (\frac{41}{4}) - 10 π}{7})$

Step 7.4.2.1.2

Simplify each term.

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Step 7.4.2.1.2.1

Cancel the common factor of $2$ .

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Step 7.4.2.1.2.1.1

Factor $2$ out of $2 π$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{2 (π) (\frac{41}{4}) - 10 π}{7})$

Step 7.4.2.1.2.1.2

Factor $2$ out of $4$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{2 (π) (\frac{41}{2 (2)}) - 10 π}{7})$

Step 7.4.2.1.2.1.3

Cancel the common factor.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{2 π (\frac{41}{2 \cdot 2}) - 10 π}{7})$

Step 7.4.2.1.2.1.4

Rewrite the expression.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{π (\frac{41}{2}) - 10 π}{7})$

Step 7.4.2.1.2.2

Combine $π$ and $\frac{41}{2}$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{π \cdot 41}{2} - 10 π}{7})$

Step 7.4.2.1.2.3

Move $41$ to the left of $π$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{41 π}{2} - 10 π}{7})$

Step 7.4.2.1.3

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

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{41 π}{2} - 10 π \cdot \frac{2}{2}}{7})$

Step 7.4.2.1.4

Combine $- 10 π$ and $\frac{2}{2}$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{41 π}{2} + \frac{- 10 π \cdot 2}{2}}{7})$

Step 7.4.2.1.5

Combine the numerators over the common denominator.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{41 π - 10 π \cdot 2}{2}}{7})$

Step 7.4.2.1.6

Simplify the numerator.

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Step 7.4.2.1.6.1

Multiply $2$ by $- 10$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{41 π - 20 π}{2}}{7})$

Step 7.4.2.1.6.2

Subtract $20 π$ from $41 π$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{\frac{21 π}{2}}{7})$

Step 7.4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{21 π}{2} \cdot \frac{1}{7})$

Step 7.4.2.1.8

Cancel the common factor of $7$ .

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Step 7.4.2.1.8.1

Factor $7$ out of $21 π$ .

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{7 (3 π)}{2} \cdot \frac{1}{7})$

Step 7.4.2.1.8.2

Cancel the common factor.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{7 (3 π)}{2} \cdot \frac{1}{7})$

Step 7.4.2.1.8.3

Rewrite the expression.

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{3 π}{2})$

Step 7.4.2.1.9

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

$f (\frac{41}{4}) = - 1 + 6 \cos (\frac{π}{2})$

Step 7.4.2.1.10

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

$f (\frac{41}{4}) = - 1 + 6 \cdot 0$

Step 7.4.2.1.11

Multiply $6$ by $0$ .

$f (\frac{41}{4}) = - 1 + 0$

Step 7.4.2.2

Add $- 1$ and $0$ .

$f (\frac{41}{4}) = - 1$

Step 7.4.2.3

The final answer is $- 1$ .

$- 1$

Step 7.5

Find the point at $x = 12$ .

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Step 7.5.1

Replace the variable $x$ with $12$ in the expression.

$f (12) = - 1 + 6 \cos (\frac{2 π (12)}{7} - \frac{10 π}{7})$

Step 7.5.2

Simplify the result.

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Step 7.5.2.1

Simplify each term.

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Step 7.5.2.1.1

Combine the numerators over the common denominator.

$f (12) = - 1 + 6 \cos (\frac{2 π (12) - 10 π}{7})$

Step 7.5.2.1.2

Multiply $12$ by $2$ .

$f (12) = - 1 + 6 \cos (\frac{24 π - 10 π}{7})$

Step 7.5.2.1.3

Subtract $10 π$ from $24 π$ .

$f (12) = - 1 + 6 \cos (\frac{14 π}{7})$

Step 7.5.2.1.4

Cancel the common factor of $14$ and $7$ .

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Step 7.5.2.1.4.1

Factor $7$ out of $14 π$ .

$f (12) = - 1 + 6 \cos (\frac{7 (2 π)}{7})$

Step 7.5.2.1.4.2

Cancel the common factors.

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Step 7.5.2.1.4.2.1

Factor $7$ out of $7$ .

$f (12) = - 1 + 6 \cos (\frac{7 (2 π)}{7 (1)})$

Step 7.5.2.1.4.2.2

Cancel the common factor.

$f (12) = - 1 + 6 \cos (\frac{7 (2 π)}{7 \cdot 1})$

Step 7.5.2.1.4.2.3

Rewrite the expression.

$f (12) = - 1 + 6 \cos (\frac{2 π}{1})$

Step 7.5.2.1.4.2.4

Divide $2 π$ by $1$ .

$f (12) = - 1 + 6 \cos (2 π)$

Step 7.5.2.1.5

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

$f (12) = - 1 + 6 \cos (0)$

Step 7.5.2.1.6

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

$f (12) = - 1 + 6 \cdot 1$

Step 7.5.2.1.7

Multiply $6$ by $1$ .

$f (12) = - 1 + 6$

Step 7.5.2.2

Add $- 1$ and $6$ .

$f (12) = 5$

Step 7.5.2.3

The final answer is $5$ .

$5$

Step 7.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 5 & 5 \\ \frac{27}{4} & - 1 \\ \frac{17}{2} & - 7 \\ \frac{41}{4} & - 1 \\ 12 & 5 \end{array}$

Step 8

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

Amplitude: $6$

Period: $7$

Phase Shift: $5$ ( $5$ to the right)

Vertical Shift: $- 1$

$\begin{array}{c} x & f (x) \\ 5 & 5 \\ \frac{27}{4} & - 1 \\ \frac{17}{2} & - 7 \\ \frac{41}{4} & - 1 \\ 12 & 5 \end{array}$

Step 9