Trigonometry Examples

$f (x) = \cos (\frac{π}{6} x + \frac{7 π}{6})$

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.

$a = 1$

$b = \frac{π}{6}$

$c = - \frac{7 π}{6}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

Tap for more steps...

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 $\frac{π}{6}$ in the formula for period.

$\frac{2 π}{| \frac{π}{6} |}$

Step 3.3

$\frac{π}{6}$ is approximately $0.52359877$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{6}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{6}{π}$

Step 3.5

Cancel the common factor of $π$ .

Tap for more steps...

Step 3.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{6}{π}$

Step 3.5.2

Cancel the common factor.

$π \cdot 2 \frac{6}{π}$

Step 3.5.3

Rewrite the expression.

$2 \cdot 6$

Step 3.6

Multiply $2$ by $6$ .

$12$

Step 4

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

Tap for more steps...

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: $\frac{- \frac{7 π}{6}}{\frac{π}{6}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{7 π}{6} \cdot \frac{6}{π}$

Step 4.4

Cancel the common factor of $π$ .

Tap for more steps...

Step 4.4.1

Move the leading negative in $- \frac{7 π}{6}$ into the numerator.

Phase Shift: $\frac{- 7 π}{6} \cdot \frac{6}{π}$

Step 4.4.2

Factor $π$ out of $- 7 π$ .

Phase Shift: $\frac{π \cdot - 7}{6} \cdot \frac{6}{π}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 7}{6} \cdot \frac{6}{π}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- 7}{6} \cdot 6$

Step 4.5

Cancel the common factor of $6$ .

Tap for more steps...

Step 4.5.1

Cancel the common factor.

Phase Shift: $\frac{- 7}{6} \cdot 6$

Step 4.5.2

Rewrite the expression.

Phase Shift: $- 7$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $12$

Phase Shift: $- 7$ ( $7$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

Tap for more steps...

Step 6.1

Find the point at $x = - 7$ .

Tap for more steps...

Step 6.1.1

Replace the variable $x$ with $- 7$ in the expression.

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

Step 6.1.2

Simplify the result.

Tap for more steps...

Step 6.1.2.1

Combine the numerators over the common denominator.

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

Step 6.1.2.2

Move $- 7$ to the left of $π$ .

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

Step 6.1.2.3

Simplify by adding terms.

Tap for more steps...

Step 6.1.2.3.1

Add $- 7 π$ and $7 π$ .

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

Step 6.1.2.3.2

Divide $0$ by $6$ .

$f (- 7) = \cos (0)$

Step 6.1.2.4

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

$f (- 7) = 1$

Step 6.1.2.5

The final answer is $1$ .

$1$

Step 6.2

Find the point at $x = - 4$ .

Tap for more steps...

Step 6.2.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 6.2.2

Simplify the result.

Tap for more steps...

Step 6.2.2.1

Combine the numerators over the common denominator.

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

Step 6.2.2.2

Move $- 4$ to the left of $π$ .

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

Step 6.2.2.3

Simplify terms.

Tap for more steps...

Step 6.2.2.3.1

Add $- 4 π$ and $7 π$ .

$f (- 4) = \cos (\frac{3 π}{6})$

Step 6.2.2.3.2

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

Tap for more steps...

Step 6.2.2.3.2.1

Factor $3$ out of $3 π$ .

$f (- 4) = \cos (\frac{3 (π)}{6})$

Step 6.2.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.3.2.2.1

Factor $3$ out of $6$ .

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

Step 6.2.2.3.2.2.2

Cancel the common factor.

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

Step 6.2.2.3.2.2.3

Rewrite the expression.

$f (- 4) = \cos (\frac{π}{2})$

Step 6.2.2.4

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

$f (- 4) = 0$

Step 6.2.2.5

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = - 1$ .

Tap for more steps...

Step 6.3.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 6.3.2

Simplify the result.

Tap for more steps...

Step 6.3.2.1

Combine the numerators over the common denominator.

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

Step 6.3.2.2

Simplify each term.

Tap for more steps...

Step 6.3.2.2.1

Move $- 1$ to the left of $π$ .

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

Step 6.3.2.2.2

Rewrite $- 1 π$ as $- π$ .

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

Step 6.3.2.3

Simplify terms.

Tap for more steps...

Step 6.3.2.3.1

Add $- π$ and $7 π$ .

$f (- 1) = \cos (\frac{6 π}{6})$

Step 6.3.2.3.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 6.3.2.3.2.1

Cancel the common factor.

$f (- 1) = \cos (\frac{6 π}{6})$

Step 6.3.2.3.2.2

Divide $π$ by $1$ .

$f (- 1) = \cos (π)$

Step 6.3.2.4

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.

$f (- 1) = - \cos (0)$

Step 6.3.2.5

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

$f (- 1) = - 1 \cdot 1$

Step 6.3.2.6

Multiply $- 1$ by $1$ .

$f (- 1) = - 1$

Step 6.3.2.7

The final answer is $- 1$ .

$- 1$

Step 6.4

Find the point at $x = 2$ .

Tap for more steps...

Step 6.4.1

Replace the variable $x$ with $2$ in the expression.

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

Step 6.4.2

Simplify the result.

Tap for more steps...

Step 6.4.2.1

Combine the numerators over the common denominator.

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

Step 6.4.2.2

Move $2$ to the left of $π$ .

$f (2) = \cos (\frac{2 π + 7 π}{6})$

Step 6.4.2.3

Simplify terms.

Tap for more steps...

Step 6.4.2.3.1

Add $2 π$ and $7 π$ .

$f (2) = \cos (\frac{9 π}{6})$

Step 6.4.2.3.2

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

Tap for more steps...

Step 6.4.2.3.2.1

Factor $3$ out of $9 π$ .

$f (2) = \cos (\frac{3 (3 π)}{6})$

Step 6.4.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.4.2.3.2.2.1

Factor $3$ out of $6$ .

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

Step 6.4.2.3.2.2.2

Cancel the common factor.

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

Step 6.4.2.3.2.2.3

Rewrite the expression.

$f (2) = \cos (\frac{3 π}{2})$

Step 6.4.2.4

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

$f (2) = \cos (\frac{π}{2})$

Step 6.4.2.5

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

$f (2) = 0$

Step 6.4.2.6

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = 5$ .

Tap for more steps...

Step 6.5.1

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

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

Step 6.5.2

Simplify the result.

Tap for more steps...

Step 6.5.2.1

Combine the numerators over the common denominator.

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

Step 6.5.2.2

Move $5$ to the left of $π$ .

$f (5) = \cos (\frac{5 π + 7 π}{6})$

Step 6.5.2.3

Simplify terms.

Tap for more steps...

Step 6.5.2.3.1

Add $5 π$ and $7 π$ .

$f (5) = \cos (\frac{12 π}{6})$

Step 6.5.2.3.2

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

Tap for more steps...

Step 6.5.2.3.2.1

Factor $6$ out of $12 π$ .

$f (5) = \cos (\frac{6 (2 π)}{6})$

Step 6.5.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.5.2.3.2.2.1

Factor $6$ out of $6$ .

$f (5) = \cos (\frac{6 (2 π)}{6 (1)})$

Step 6.5.2.3.2.2.2

Cancel the common factor.

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

Step 6.5.2.3.2.2.3

Rewrite the expression.

$f (5) = \cos (\frac{2 π}{1})$

Step 6.5.2.3.2.2.4

Divide $2 π$ by $1$ .

$f (5) = \cos (2 π)$

Step 6.5.2.4

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

$f (5) = \cos (0)$

Step 6.5.2.5

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

$f (5) = 1$

Step 6.5.2.6

The final answer is $1$ .

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 7 & 1 \\ - 4 & 0 \\ - 1 & - 1 \\ 2 & 0 \\ 5 & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $12$

Phase Shift: $- 7$ ( $7$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 7 & 1 \\ - 4 & 0 \\ - 1 & - 1 \\ 2 & 0 \\ 5 & 1 \end{array}$

Step 8