Trigonometry Examples

$y = 2 \cos (x + \frac{π}{6}) + 2$

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.

$a = 2$

$b = 1$

$c = - \frac{π}{6}$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

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

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Step 3.1

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.1.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2

Find the period of $2$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.2.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.3

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

$2 π$

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: $\frac{- \frac{π}{6}}{1}$

Step 4.3

Divide $- \frac{π}{6}$ by $1$ .

Phase Shift: $- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $2 π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: $2$

Step 6

Select a few points to graph.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Combine the numerators over the common denominator.

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

Step 6.1.2.1.2

Add $- π$ and $π$ .

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

Step 6.1.2.1.3

Divide $0$ by $6$ .

$f (- \frac{π}{6}) = 2 \cos (0) + 2$

Step 6.1.2.1.4

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

$f (- \frac{π}{6}) = 2 \cdot 1 + 2$

Step 6.1.2.1.5

Multiply $2$ by $1$ .

$f (- \frac{π}{6}) = 2 + 2$

Step 6.1.2.2

Add $2$ and $2$ .

$f (- \frac{π}{6}) = 4$

Step 6.1.2.3

The final answer is $4$ .

$4$

Step 6.2

Find the point at $x = \frac{π}{3}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{π}{3}$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.2.1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.1.2.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

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

Step 6.2.2.1.2.2

Multiply $3$ by $2$ .

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

Step 6.2.2.1.3

Combine the numerators over the common denominator.

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

Step 6.2.2.1.4

Simplify the numerator.

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Step 6.2.2.1.4.1

Move $2$ to the left of $π$ .

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

Step 6.2.2.1.4.2

Add $2 π$ and $π$ .

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

Step 6.2.2.1.5

Cancel the common factor of $3$ and $6$ .

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Step 6.2.2.1.5.1

Factor $3$ out of $3 π$ .

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

Step 6.2.2.1.5.2

Cancel the common factors.

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Step 6.2.2.1.5.2.1

Factor $3$ out of $6$ .

$f (\frac{π}{3}) = 2 \cos (\frac{3 π}{3 \cdot 2}) + 2$

Step 6.2.2.1.5.2.2

Cancel the common factor.

$f (\frac{π}{3}) = 2 \cos (\frac{3 π}{3 \cdot 2}) + 2$

Step 6.2.2.1.5.2.3

Rewrite the expression.

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

Step 6.2.2.1.6

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

$f (\frac{π}{3}) = 2 \cdot 0 + 2$

Step 6.2.2.1.7

Multiply $2$ by $0$ .

$f (\frac{π}{3}) = 0 + 2$

Step 6.2.2.2

Add $0$ and $2$ .

$f (\frac{π}{3}) = 2$

Step 6.2.2.3

The final answer is $2$ .

$2$

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Combine the numerators over the common denominator.

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

Step 6.3.2.1.2

Add $5 π$ and $π$ .

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

Step 6.3.2.1.3

Cancel the common factor of $6$ .

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Step 6.3.2.1.3.1

Cancel the common factor.

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

Step 6.3.2.1.3.2

Divide $π$ by $1$ .

$f (\frac{5 π}{6}) = 2 \cos (π) + 2$

Step 6.3.2.1.4

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.

$f (\frac{5 π}{6}) = 2 (- \cos (0)) + 2$

Step 6.3.2.1.5

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

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

Step 6.3.2.1.6

Multiply $2 (- 1 \cdot 1)$ .

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Step 6.3.2.1.6.1

Multiply $- 1$ by $1$ .

$f (\frac{5 π}{6}) = 2 \cdot - 1 + 2$

Step 6.3.2.1.6.2

Multiply $2$ by $- 1$ .

$f (\frac{5 π}{6}) = - 2 + 2$

Step 6.3.2.2

Add $- 2$ and $2$ .

$f (\frac{5 π}{6}) = 0$

Step 6.3.2.3

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{4 π}{3}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{4 π}{3}$ in the expression.

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.4.2.1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.1.2.1

Multiply $\frac{4 π}{3}$ by $\frac{2}{2}$ .

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

Step 6.4.2.1.2.2

Multiply $3$ by $2$ .

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

Step 6.4.2.1.3

Combine the numerators over the common denominator.

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

Step 6.4.2.1.4

Simplify the numerator.

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Step 6.4.2.1.4.1

Multiply $2$ by $4$ .

$f (\frac{4 π}{3}) = 2 \cos (\frac{8 π + π}{6}) + 2$

Step 6.4.2.1.4.2

Add $8 π$ and $π$ .

$f (\frac{4 π}{3}) = 2 \cos (\frac{9 π}{6}) + 2$

Step 6.4.2.1.5

Cancel the common factor of $9$ and $6$ .

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Step 6.4.2.1.5.1

Factor $3$ out of $9 π$ .

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

Step 6.4.2.1.5.2

Cancel the common factors.

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Step 6.4.2.1.5.2.1

Factor $3$ out of $6$ .

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

Step 6.4.2.1.5.2.2

Cancel the common factor.

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

Step 6.4.2.1.5.2.3

Rewrite the expression.

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

Step 6.4.2.1.6

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

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

Step 6.4.2.1.7

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

$f (\frac{4 π}{3}) = 2 \cdot 0 + 2$

Step 6.4.2.1.8

Multiply $2$ by $0$ .

$f (\frac{4 π}{3}) = 0 + 2$

Step 6.4.2.2

Add $0$ and $2$ .

$f (\frac{4 π}{3}) = 2$

Step 6.4.2.3

The final answer is $2$ .

$2$

Step 6.5

Find the point at $x = \frac{11 π}{6}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{11 π}{6}$ in the expression.

$f (\frac{11 π}{6}) = 2 \cos ((\frac{11 π}{6}) + \frac{π}{6}) + 2$

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

Combine the numerators over the common denominator.

$f (\frac{11 π}{6}) = 2 \cos (\frac{11 π + π}{6}) + 2$

Step 6.5.2.1.2

Add $11 π$ and $π$ .

$f (\frac{11 π}{6}) = 2 \cos (\frac{12 π}{6}) + 2$

Step 6.5.2.1.3

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

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Step 6.5.2.1.3.1

Factor $6$ out of $12 π$ .

$f (\frac{11 π}{6}) = 2 \cos (\frac{6 (2 π)}{6}) + 2$

Step 6.5.2.1.3.2

Cancel the common factors.

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Step 6.5.2.1.3.2.1

Factor $6$ out of $6$ .

$f (\frac{11 π}{6}) = 2 \cos (\frac{6 (2 π)}{6 (1)}) + 2$

Step 6.5.2.1.3.2.2

Cancel the common factor.

$f (\frac{11 π}{6}) = 2 \cos (\frac{6 (2 π)}{6 \cdot 1}) + 2$

Step 6.5.2.1.3.2.3

Rewrite the expression.

$f (\frac{11 π}{6}) = 2 \cos (\frac{2 π}{1}) + 2$

Step 6.5.2.1.3.2.4

Divide $2 π$ by $1$ .

$f (\frac{11 π}{6}) = 2 \cos (2 π) + 2$

Step 6.5.2.1.4

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

$f (\frac{11 π}{6}) = 2 \cos (0) + 2$

Step 6.5.2.1.5

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

$f (\frac{11 π}{6}) = 2 \cdot 1 + 2$

Step 6.5.2.1.6

Multiply $2$ by $1$ .

$f (\frac{11 π}{6}) = 2 + 2$

Step 6.5.2.2

Add $2$ and $2$ .

$f (\frac{11 π}{6}) = 4$

Step 6.5.2.3

The final answer is $4$ .

$4$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{6} & 4 \\ \frac{π}{3} & 2 \\ \frac{5 π}{6} & 0 \\ \frac{4 π}{3} & 2 \\ \frac{11 π}{6} & 4 \end{array}$

Step 7

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

Amplitude: $2$

Period: $2 π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ - \frac{π}{6} & 4 \\ \frac{π}{3} & 2 \\ \frac{5 π}{6} & 0 \\ \frac{4 π}{3} & 2 \\ \frac{11 π}{6} & 4 \end{array}$

Step 8