Trigonometry Examples

$y = \frac{1}{4} \cdot \cos (\frac{1}{3} x - \frac{π}{12})$

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.

$a = \frac{1}{4}$

$b = \frac{1}{3}$

$c = \frac{π}{12}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

Step 3

Find the period of $\frac{\cos (\frac{x}{3} - \frac{π}{12})}{4}$ .

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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 $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

Step 3.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Step 3.5

Multiply $3$ by $2$ .

$6 π$

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: $\frac{\frac{π}{12}}{\frac{1}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{12} \cdot 3$

Step 4.4

Cancel the common factor of $3$ .

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Step 4.4.1

Factor $3$ out of $12$ .

Phase Shift: $\frac{π}{3 (4)} \cdot 3$

Step 4.4.2

Cancel the common factor.

Phase Shift: $\frac{π}{3 \cdot 4} \cdot 3$

Step 4.4.3

Rewrite the expression.

Phase Shift: $\frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $6 π$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ 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 $x = \frac{π}{4}$ .

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Step 6.1.1

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

$f (\frac{π}{4}) = \frac{\cos (\frac{\frac{π}{4}}{3} - \frac{π}{12})}{4}$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify the numerator.

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Step 6.1.2.1.1

Simplify each term.

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Step 6.1.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{π}{4}) = \frac{\cos (\frac{π}{4} \cdot \frac{1}{3} - \frac{π}{12})}{4}$

Step 6.1.2.1.1.2

Multiply $\frac{π}{4} \cdot \frac{1}{3}$ .

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Step 6.1.2.1.1.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{3}$ .

$f (\frac{π}{4}) = \frac{\cos (\frac{π}{4 \cdot 3} - \frac{π}{12})}{4}$

Step 6.1.2.1.1.2.2

Multiply $4$ by $3$ .

$f (\frac{π}{4}) = \frac{\cos (\frac{π}{12} - \frac{π}{12})}{4}$

Step 6.1.2.1.2

Combine the numerators over the common denominator.

$f (\frac{π}{4}) = \frac{\cos (\frac{π - π}{12})}{4}$

Step 6.1.2.1.3

Subtract $π$ from $π$ .

$f (\frac{π}{4}) = \frac{\cos (\frac{0}{12})}{4}$

Step 6.1.2.1.4

Divide $0$ by $12$ .

$f (\frac{π}{4}) = \frac{\cos (0)}{4}$

Step 6.1.2.1.5

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

$f (\frac{π}{4}) = \frac{1}{4}$

Step 6.1.2.2

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 6.2

Find the point at $x = \frac{7 π}{4}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{7 π}{4}$ in the expression.

$f (\frac{7 π}{4}) = \frac{\cos (\frac{\frac{7 π}{4}}{3} - \frac{π}{12})}{4}$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify the numerator.

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Step 6.2.2.1.1

Simplify each term.

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Step 6.2.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 π}{4}) = \frac{\cos (\frac{7 π}{4} \cdot \frac{1}{3} - \frac{π}{12})}{4}$

Step 6.2.2.1.1.2

Multiply $\frac{7 π}{4} \cdot \frac{1}{3}$ .

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Step 6.2.2.1.1.2.1

Multiply $\frac{7 π}{4}$ by $\frac{1}{3}$ .

$f (\frac{7 π}{4}) = \frac{\cos (\frac{7 π}{4 \cdot 3} - \frac{π}{12})}{4}$

Step 6.2.2.1.1.2.2

Multiply $4$ by $3$ .

$f (\frac{7 π}{4}) = \frac{\cos (\frac{7 π}{12} - \frac{π}{12})}{4}$

Step 6.2.2.1.2

Combine the numerators over the common denominator.

$f (\frac{7 π}{4}) = \frac{\cos (\frac{7 π - π}{12})}{4}$

Step 6.2.2.1.3

Subtract $π$ from $7 π$ .

$f (\frac{7 π}{4}) = \frac{\cos (\frac{6 π}{12})}{4}$

Step 6.2.2.1.4

Cancel the common factor of $6$ and $12$ .

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Step 6.2.2.1.4.1

Factor $6$ out of $6 π$ .

$f (\frac{7 π}{4}) = \frac{\cos (\frac{6 (π)}{12})}{4}$

Step 6.2.2.1.4.2

Cancel the common factors.

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Step 6.2.2.1.4.2.1

Factor $6$ out of $12$ .

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

Step 6.2.2.1.4.2.2

Cancel the common factor.

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

Step 6.2.2.1.4.2.3

Rewrite the expression.

$f (\frac{7 π}{4}) = \frac{\cos (\frac{π}{2})}{4}$

Step 6.2.2.1.5

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

$f (\frac{7 π}{4}) = \frac{0}{4}$

Step 6.2.2.2

Divide $0$ by $4$ .

$f (\frac{7 π}{4}) = 0$

Step 6.2.2.3

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{13 π}{4}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{13 π}{4}$ in the expression.

$f (\frac{13 π}{4}) = \frac{\cos (\frac{\frac{13 π}{4}}{3} - \frac{π}{12})}{4}$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify the numerator.

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Step 6.3.2.1.1

Simplify each term.

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Step 6.3.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{13 π}{4}) = \frac{\cos (\frac{13 π}{4} \cdot \frac{1}{3} - \frac{π}{12})}{4}$

Step 6.3.2.1.1.2

Multiply $\frac{13 π}{4} \cdot \frac{1}{3}$ .

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Step 6.3.2.1.1.2.1

Multiply $\frac{13 π}{4}$ by $\frac{1}{3}$ .

$f (\frac{13 π}{4}) = \frac{\cos (\frac{13 π}{4 \cdot 3} - \frac{π}{12})}{4}$

Step 6.3.2.1.1.2.2

Multiply $4$ by $3$ .

$f (\frac{13 π}{4}) = \frac{\cos (\frac{13 π}{12} - \frac{π}{12})}{4}$

Step 6.3.2.1.2

Combine the numerators over the common denominator.

$f (\frac{13 π}{4}) = \frac{\cos (\frac{13 π - π}{12})}{4}$

Step 6.3.2.1.3

Subtract $π$ from $13 π$ .

$f (\frac{13 π}{4}) = \frac{\cos (\frac{12 π}{12})}{4}$

Step 6.3.2.1.4

Cancel the common factor of $12$ .

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Step 6.3.2.1.4.1

Cancel the common factor.

$f (\frac{13 π}{4}) = \frac{\cos (\frac{12 π}{12})}{4}$

Step 6.3.2.1.4.2

Divide $π$ by $1$ .

$f (\frac{13 π}{4}) = \frac{\cos (π)}{4}$

Step 6.3.2.1.5

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.

$f (\frac{13 π}{4}) = \frac{- \cos (0)}{4}$

Step 6.3.2.1.6

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

$f (\frac{13 π}{4}) = \frac{- 1 \cdot 1}{4}$

Step 6.3.2.1.7

Multiply $- 1$ by $1$ .

$f (\frac{13 π}{4}) = \frac{- 1}{4}$

Step 6.3.2.2

Move the negative in front of the fraction.

$f (\frac{13 π}{4}) = - \frac{1}{4}$

Step 6.3.2.3

The final answer is $- \frac{1}{4}$ .

$- \frac{1}{4}$

Step 6.4

Find the point at $x = \frac{19 π}{4}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{19 π}{4}$ in the expression.

$f (\frac{19 π}{4}) = \frac{\cos (\frac{\frac{19 π}{4}}{3} - \frac{π}{12})}{4}$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify the numerator.

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Step 6.4.2.1.1

Simplify each term.

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Step 6.4.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{19 π}{4}) = \frac{\cos (\frac{19 π}{4} \cdot \frac{1}{3} - \frac{π}{12})}{4}$

Step 6.4.2.1.1.2

Multiply $\frac{19 π}{4} \cdot \frac{1}{3}$ .

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Step 6.4.2.1.1.2.1

Multiply $\frac{19 π}{4}$ by $\frac{1}{3}$ .

$f (\frac{19 π}{4}) = \frac{\cos (\frac{19 π}{4 \cdot 3} - \frac{π}{12})}{4}$

Step 6.4.2.1.1.2.2

Multiply $4$ by $3$ .

$f (\frac{19 π}{4}) = \frac{\cos (\frac{19 π}{12} - \frac{π}{12})}{4}$

Step 6.4.2.1.2

Combine the numerators over the common denominator.

$f (\frac{19 π}{4}) = \frac{\cos (\frac{19 π - π}{12})}{4}$

Step 6.4.2.1.3

Subtract $π$ from $19 π$ .

$f (\frac{19 π}{4}) = \frac{\cos (\frac{18 π}{12})}{4}$

Step 6.4.2.1.4

Cancel the common factor of $18$ and $12$ .

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Step 6.4.2.1.4.1

Factor $6$ out of $18 π$ .

$f (\frac{19 π}{4}) = \frac{\cos (\frac{6 (3 π)}{12})}{4}$

Step 6.4.2.1.4.2

Cancel the common factors.

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Step 6.4.2.1.4.2.1

Factor $6$ out of $12$ .

$f (\frac{19 π}{4}) = \frac{\cos (\frac{6 (3 π)}{6 (2)})}{4}$

Step 6.4.2.1.4.2.2

Cancel the common factor.

$f (\frac{19 π}{4}) = \frac{\cos (\frac{6 (3 π)}{6 \cdot 2})}{4}$

Step 6.4.2.1.4.2.3

Rewrite the expression.

$f (\frac{19 π}{4}) = \frac{\cos (\frac{3 π}{2})}{4}$

Step 6.4.2.1.5

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

$f (\frac{19 π}{4}) = \frac{\cos (\frac{π}{2})}{4}$

Step 6.4.2.1.6

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

$f (\frac{19 π}{4}) = \frac{0}{4}$

Step 6.4.2.2

Divide $0$ by $4$ .

$f (\frac{19 π}{4}) = 0$

Step 6.4.2.3

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{25 π}{4}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{25 π}{4}$ in the expression.

$f (\frac{25 π}{4}) = \frac{\cos (\frac{\frac{25 π}{4}}{3} - \frac{π}{12})}{4}$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify the numerator.

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Step 6.5.2.1.1

Simplify each term.

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Step 6.5.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{25 π}{4}) = \frac{\cos (\frac{25 π}{4} \cdot \frac{1}{3} - \frac{π}{12})}{4}$

Step 6.5.2.1.1.2

Multiply $\frac{25 π}{4} \cdot \frac{1}{3}$ .

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Step 6.5.2.1.1.2.1

Multiply $\frac{25 π}{4}$ by $\frac{1}{3}$ .

$f (\frac{25 π}{4}) = \frac{\cos (\frac{25 π}{4 \cdot 3} - \frac{π}{12})}{4}$

Step 6.5.2.1.1.2.2

Multiply $4$ by $3$ .

$f (\frac{25 π}{4}) = \frac{\cos (\frac{25 π}{12} - \frac{π}{12})}{4}$

Step 6.5.2.1.2

Combine the numerators over the common denominator.

$f (\frac{25 π}{4}) = \frac{\cos (\frac{25 π - π}{12})}{4}$

Step 6.5.2.1.3

Subtract $π$ from $25 π$ .

$f (\frac{25 π}{4}) = \frac{\cos (\frac{24 π}{12})}{4}$

Step 6.5.2.1.4

Cancel the common factor of $24$ and $12$ .

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Step 6.5.2.1.4.1

Factor $12$ out of $24 π$ .

$f (\frac{25 π}{4}) = \frac{\cos (\frac{12 (2 π)}{12})}{4}$

Step 6.5.2.1.4.2

Cancel the common factors.

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Step 6.5.2.1.4.2.1

Factor $12$ out of $12$ .

$f (\frac{25 π}{4}) = \frac{\cos (\frac{12 (2 π)}{12 (1)})}{4}$

Step 6.5.2.1.4.2.2

Cancel the common factor.

$f (\frac{25 π}{4}) = \frac{\cos (\frac{12 (2 π)}{12 \cdot 1})}{4}$

Step 6.5.2.1.4.2.3

Rewrite the expression.

$f (\frac{25 π}{4}) = \frac{\cos (\frac{2 π}{1})}{4}$

Step 6.5.2.1.4.2.4

Divide $2 π$ by $1$ .

$f (\frac{25 π}{4}) = \frac{\cos (2 π)}{4}$

Step 6.5.2.1.5

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

$f (\frac{25 π}{4}) = \frac{\cos (0)}{4}$

Step 6.5.2.1.6

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

$f (\frac{25 π}{4}) = \frac{1}{4}$

Step 6.5.2.2

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{4} & \frac{1}{4} \\ \frac{7 π}{4} & 0 \\ \frac{13 π}{4} & - \frac{1}{4} \\ \frac{19 π}{4} & 0 \\ \frac{25 π}{4} & \frac{1}{4} \end{array}$

Step 7

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

Amplitude: $\frac{1}{4}$

Period: $6 π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{π}{4} & \frac{1}{4} \\ \frac{7 π}{4} & 0 \\ \frac{13 π}{4} & - \frac{1}{4} \\ \frac{19 π}{4} & 0 \\ \frac{25 π}{4} & \frac{1}{4} \end{array}$

Step 8