Trigonometry Examples

$\frac{1}{9} \cdot \arccos (x - \frac{π}{6})$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - 1 + \frac{π}{6}$ .

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Step 1.1.1

Replace the variable $x$ with $- 1 + \frac{π}{6}$ in the expression.

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{6 (- 1 + \frac{π}{6}) - π}{6})}{9}$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Simplify the numerator.

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Step 1.1.2.1.1

Apply the distributive property.

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{6 \cdot - 1 + 6 (\frac{π}{6}) - π}{6})}{9}$

Step 1.1.2.1.2

Multiply $6$ by $- 1$ .

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{- 6 + 6 (\frac{π}{6}) - π}{6})}{9}$

Step 1.1.2.1.3

Cancel the common factor of $6$ .

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Step 1.1.2.1.3.1

Cancel the common factor.

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{- 6 + 6 (\frac{π}{6}) - π}{6})}{9}$

Step 1.1.2.1.3.2

Rewrite the expression.

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{- 6 + π - π}{6})}{9}$

Step 1.1.2.1.4

Subtract $π$ from $π$ .

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{- 6 + 0}{6})}{9}$

Step 1.1.2.1.5

Add $- 6$ and $0$ .

$f (- 1 + \frac{π}{6}) = \frac{\arccos (\frac{- 6}{6})}{9}$

Step 1.1.2.2

Simplify the numerator.

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Step 1.1.2.2.1

Divide $- 6$ by $6$ .

$f (- 1 + \frac{π}{6}) = \frac{\arccos (- 1)}{9}$

Step 1.1.2.2.2

The exact value of $\arccos (- 1)$ is $π$ .

$f (- 1 + \frac{π}{6}) = \frac{π}{9}$

Step 1.1.2.3

The final answer is $\frac{π}{9}$ .

$\frac{π}{9}$

Step 1.2

Find the point at $x = - \frac{2 - π}{4}$ .

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Step 1.2.1

Replace the variable $x$ with $- \frac{2 - π}{4}$ in the expression.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{6 (- \frac{2 - π}{4}) - π}{6})}{9}$

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

Simplify the numerator.

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Step 1.2.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1.1

Move the leading negative in $- \frac{2 - π}{4}$ into the numerator.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{6 (\frac{- (2 - π)}{4}) - π}{6})}{9}$

Step 1.2.2.1.1.2

Factor $2$ out of $6$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{2 (3) (\frac{- (2 - π)}{4}) - π}{6})}{9}$

Step 1.2.2.1.1.3

Factor $2$ out of $4$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{2 \cdot (3 (\frac{- (2 - π)}{2 \cdot 2})) - π}{6})}{9}$

Step 1.2.2.1.1.4

Cancel the common factor.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{2 \cdot (3 (\frac{- (2 - π)}{2 \cdot 2})) - π}{6})}{9}$

Step 1.2.2.1.1.5

Rewrite the expression.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{3 (\frac{- (2 - π)}{2}) - π}{6})}{9}$

Step 1.2.2.1.2

Combine $3$ and $\frac{- (2 - π)}{2}$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{\frac{3 (- (2 - π))}{2} - π}{6})}{9}$

Step 1.2.2.1.3

Multiply $- 1$ by $3$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{\frac{- 3 (2 - π)}{2} - π}{6})}{9}$

Step 1.2.2.1.4

Move the negative in front of the fraction.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{- \frac{3 (2 - π)}{2} - π}{6})}{9}$

Step 1.2.2.1.5

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

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{- \frac{3 (2 - π)}{2} - π \cdot \frac{2}{2}}{6})}{9}$

Step 1.2.2.1.6

Combine $- π$ and $\frac{2}{2}$ .

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

Step 1.2.2.1.7

Combine the numerators over the common denominator.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{\frac{- 3 (2 - π) - π \cdot 2}{2}}{6})}{9}$

Step 1.2.2.1.8

Rewrite $\frac{- 3 (2 - π) - π \cdot 2}{2}$ in a factored form.

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Step 1.2.2.1.8.1

Apply the distributive property.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{\frac{- 3 \cdot 2 - 3 (- π) - π \cdot 2}{2}}{6})}{9}$

Step 1.2.2.1.8.2

Multiply $- 3$ by $2$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{\frac{- 6 - 3 (- π) - π \cdot 2}{2}}{6})}{9}$

Step 1.2.2.1.8.3

Multiply $- 1$ by $- 3$ .

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

Step 1.2.2.1.8.4

Multiply $2$ by $- 1$ .

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

Step 1.2.2.1.8.5

Subtract $2 π$ from $3 π$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{\frac{- 6 + π}{2}}{6})}{9}$

Step 1.2.2.2

Simplify the numerator.

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Step 1.2.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{- 6 + π}{2} \cdot \frac{1}{6})}{9}$

Step 1.2.2.2.2

Multiply $\frac{- 6 + π}{2} \cdot \frac{1}{6}$ .

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Step 1.2.2.2.2.1

Multiply $\frac{- 6 + π}{2}$ by $\frac{1}{6}$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{- 6 + π}{2 \cdot 6})}{9}$

Step 1.2.2.2.2.2

Multiply $2$ by $6$ .

$f (- \frac{2 - π}{4}) = \frac{\arccos (\frac{- 6 + π}{12})}{9}$

Step 1.2.2.2.3

Evaluate $\arccos (\frac{- 6 + π}{12})$ .

$f (- \frac{2 - π}{4}) = \frac{1.81130903}{9}$

Step 1.2.2.3

Divide $1.81130903$ by $9$ .

$f (- \frac{2 - π}{4}) = 0.20125655$

Step 1.2.2.4

The final answer is $0.20125655$ .

$0.20125655$

Step 1.3

Find the point at $x = \frac{π}{3}$ .

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Step 1.3.1

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

$f (\frac{π}{3}) = \frac{\arccos (\frac{6 (\frac{π}{3}) - π}{6})}{9}$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Simplify the numerator.

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Step 1.3.2.1.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1.1

Factor $3$ out of $6$ .

$f (\frac{π}{3}) = \frac{\arccos (\frac{3 (2) (\frac{π}{3}) - π}{6})}{9}$

Step 1.3.2.1.1.2

Cancel the common factor.

$f (\frac{π}{3}) = \frac{\arccos (\frac{3 \cdot (2 (\frac{π}{3})) - π}{6})}{9}$

Step 1.3.2.1.1.3

Rewrite the expression.

$f (\frac{π}{3}) = \frac{\arccos (\frac{2 π - π}{6})}{9}$

Step 1.3.2.1.2

Subtract $π$ from $2 π$ .

$f (\frac{π}{3}) = \frac{\arccos (\frac{π}{6})}{9}$

Step 1.3.2.2

Evaluate $\arccos (\frac{π}{6})$ .

$f (\frac{π}{3}) = \frac{1.01972674}{9}$

Step 1.3.2.3

Divide $1.01972674$ by $9$ .

$f (\frac{π}{3}) = 0.11330297$

Step 1.3.2.4

The final answer is $0.11330297$ .

$0.11330297$

Step 1.4

Find the point at $x = \frac{6 + 5 π}{12}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{6 + 5 π}{12}$ in the expression.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{6 (\frac{6 + 5 π}{12}) - π}{6})}{9}$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

Simplify the numerator.

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Step 1.4.2.1.1

Cancel the common factor of $6$ .

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Step 1.4.2.1.1.1

Factor $6$ out of $12$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{6 (\frac{6 + 5 π}{6 (2)}) - π}{6})}{9}$

Step 1.4.2.1.1.2

Cancel the common factor.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{6 (\frac{6 + 5 π}{6 \cdot 2}) - π}{6})}{9}$

Step 1.4.2.1.1.3

Rewrite the expression.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{\frac{6 + 5 π}{2} - π}{6})}{9}$

Step 1.4.2.1.2

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

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{\frac{6 + 5 π}{2} - π \cdot \frac{2}{2}}{6})}{9}$

Step 1.4.2.1.3

Combine $- π$ and $\frac{2}{2}$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{\frac{6 + 5 π}{2} + \frac{- π \cdot 2}{2}}{6})}{9}$

Step 1.4.2.1.4

Combine the numerators over the common denominator.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{\frac{6 + 5 π - π \cdot 2}{2}}{6})}{9}$

Step 1.4.2.1.5

Rewrite $\frac{6 + 5 π - π \cdot 2}{2}$ in a factored form.

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Step 1.4.2.1.5.1

Multiply $2$ by $- 1$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{\frac{6 + 5 π - 2 π}{2}}{6})}{9}$

Step 1.4.2.1.5.2

Subtract $2 π$ from $5 π$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{\frac{6 + 3 π}{2}}{6})}{9}$

Step 1.4.2.2

Simplify the numerator.

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Step 1.4.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{6 + 3 π}{2} \cdot \frac{1}{6})}{9}$

Step 1.4.2.2.2

Multiply $\frac{6 + 3 π}{2} \cdot \frac{1}{6}$ .

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Step 1.4.2.2.2.1

Multiply $\frac{6 + 3 π}{2}$ by $\frac{1}{6}$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{6 + 3 π}{2 \cdot 6})}{9}$

Step 1.4.2.2.2.2

Multiply $2$ by $6$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{6 + 3 π}{12})}{9}$

Step 1.4.2.2.3

Cancel the common factor of $6 + 3 π$ and $12$ .

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Step 1.4.2.2.3.1

Factor $3$ out of $6$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{3 \cdot 2 + 3 π}{12})}{9}$

Step 1.4.2.2.3.2

Factor $3$ out of $3 π$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{3 \cdot 2 + 3 (π)}{12})}{9}$

Step 1.4.2.2.3.3

Factor $3$ out of $3 \cdot 2 + 3 (π)$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{3 \cdot (2 + π)}{12})}{9}$

Step 1.4.2.2.3.4

Cancel the common factors.

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Step 1.4.2.2.3.4.1

Factor $3$ out of $12$ .

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{3 \cdot (2 + π)}{3 (4)})}{9}$

Step 1.4.2.2.3.4.2

Cancel the common factor.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{3 \cdot (2 + π)}{3 \cdot 4})}{9}$

Step 1.4.2.2.3.4.3

Rewrite the expression.

$f (\frac{6 + 5 π}{12}) = \frac{\arccos (\frac{2 + π}{4})}{9}$

Step 1.4.2.3

The final answer is $\frac{\arccos (\frac{2 + π}{4})}{9}$ .

$\frac{\arccos (\frac{2 + π}{4})}{9}$

Step 1.5

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

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Step 1.5.1

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

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{6 (\frac{2 + π}{2}) - π}{6})}{9}$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Simplify the numerator.

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Step 1.5.2.1.1

Cancel the common factor of $2$ .

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Step 1.5.2.1.1.1

Factor $2$ out of $6$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 (3) (\frac{2 + π}{2}) - π}{6})}{9}$

Step 1.5.2.1.1.2

Cancel the common factor.

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 \cdot (3 (\frac{2 + π}{2})) - π}{6})}{9}$

Step 1.5.2.1.1.3

Rewrite the expression.

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{3 (2 + π) - π}{6})}{9}$

Step 1.5.2.1.2

Apply the distributive property.

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{3 \cdot 2 + 3 π - π}{6})}{9}$

Step 1.5.2.1.3

Multiply $3$ by $2$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{6 + 3 π - π}{6})}{9}$

Step 1.5.2.1.4

Subtract $π$ from $3 π$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{6 + 2 π}{6})}{9}$

Step 1.5.2.2

Cancel the common factor of $6 + 2 π$ and $6$ .

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Step 1.5.2.2.1

Factor $2$ out of $6$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 \cdot 3 + 2 π}{6})}{9}$

Step 1.5.2.2.2

Factor $2$ out of $2 π$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 \cdot 3 + 2 (π)}{6})}{9}$

Step 1.5.2.2.3

Factor $2$ out of $2 \cdot 3 + 2 (π)$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 \cdot (3 + π)}{6})}{9}$

Step 1.5.2.2.4

Cancel the common factors.

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Step 1.5.2.2.4.1

Factor $2$ out of $6$ .

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 \cdot (3 + π)}{2 (3)})}{9}$

Step 1.5.2.2.4.2

Cancel the common factor.

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{2 \cdot (3 + π)}{2 \cdot 3})}{9}$

Step 1.5.2.2.4.3

Rewrite the expression.

$f (\frac{2 + π}{2}) = \frac{\arccos (\frac{3 + π}{3})}{9}$

Step 1.5.2.3

The final answer is $\frac{\arccos (\frac{3 + π}{3})}{9}$ .

$\frac{\arccos (\frac{3 + π}{3})}{9}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 1 + \frac{π}{6} & \frac{π}{9} \\ - \frac{2 - π}{4} & 0.201 \\ \frac{π}{3} & 0.113 \\ \frac{6 + 5 π}{12} & \frac{\arccos (\frac{2 + π}{4})}{9} \\ \frac{2 + π}{2} & \frac{\arccos (\frac{3 + π}{3})}{9} \end{array}$

Step 2

The trig function can be graphed using the points.

Step 3