Trigonometry Examples

$y = 2 \arccos (x + \frac{π}{2})$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - 1 - \frac{π}{2}$ .

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Step 1.1.1

Replace the variable $x$ with $- 1 - \frac{π}{2}$ in the expression.

$f (- 1 - \frac{π}{2}) = 2 \arccos ((- 1 - \frac{π}{2}) + \frac{π}{2})$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Combine the numerators over the common denominator.

$f (- 1 - \frac{π}{2}) = 2 \arccos (- 1 + \frac{- π + π}{2})$

Step 1.1.2.2

Add $- π$ and $π$ .

$f (- 1 - \frac{π}{2}) = 2 \arccos (- 1 + \frac{0}{2})$

Step 1.1.2.3

Simplify the expression.

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Step 1.1.2.3.1

Divide $0$ by $2$ .

$f (- 1 - \frac{π}{2}) = 2 \arccos (- 1 + 0)$

Step 1.1.2.3.2

Add $- 1$ and $0$ .

$f (- 1 - \frac{π}{2}) = 2 \arccos (- 1)$

Step 1.1.2.4

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

$f (- 1 - \frac{π}{2}) = 2 π$

Step 1.1.2.5

The final answer is $2 π$ .

$2 π$

Step 1.2

Find the point at $x = - \frac{2 + 3 π}{4}$ .

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Step 1.2.1

Replace the variable $x$ with $- \frac{2 + 3 π}{4}$ in the expression.

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

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

Multiply $2$ by $2$ .

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

Step 1.2.2.3

Combine the numerators over the common denominator.

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

Step 1.2.2.4

Simplify the numerator.

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Step 1.2.2.4.1

Apply the distributive property.

$f (- \frac{2 + 3 π}{4}) = 2 \arccos (\frac{- 1 \cdot 2 - (3 π) + π \cdot 2}{4})$

Step 1.2.2.4.2

Multiply $- 1$ by $2$ .

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

Step 1.2.2.4.3

Multiply $3$ by $- 1$ .

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

Step 1.2.2.4.4

Move $2$ to the left of $π$ .

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

Step 1.2.2.4.5

Add $- 3 π$ and $2 π$ .

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

Step 1.2.2.5

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

$2 \arccos (\frac{- 2 - π}{4})$

Step 1.3

Find the point at $x = - π$ .

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Step 1.3.1

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

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

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Combine fractions.

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Step 1.3.2.2.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 1.3.2.2.2

Combine the numerators over the common denominator.

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

Step 1.3.2.3

Simplify the numerator.

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Step 1.3.2.3.1

Multiply $2$ by $- 1$ .

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

Step 1.3.2.3.2

Add $- 2 π$ and $π$ .

$f (- π) = 2 \arccos (\frac{- π}{2})$

Step 1.3.2.4

Move the negative in front of the fraction.

$f (- π) = 2 \arccos (- \frac{π}{2})$

Step 1.3.2.5

The final answer is $2 \arccos (- \frac{π}{2})$ .

$2 \arccos (- \frac{π}{2})$

Step 1.4

Find the point at $x = \frac{2 - 5 π}{4}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{2 - 5 π}{4}$ in the expression.

$f (\frac{2 - 5 π}{4}) = 2 \arccos ((\frac{2 - 5 π}{4}) + \frac{π}{2})$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

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

$f (\frac{2 - 5 π}{4}) = 2 \arccos (\frac{2 - 5 π}{4} + \frac{π}{2} \cdot \frac{2}{2})$

Step 1.4.2.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.2.2.1

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

$f (\frac{2 - 5 π}{4}) = 2 \arccos (\frac{2 - 5 π}{4} + \frac{π \cdot 2}{2 \cdot 2})$

Step 1.4.2.2.2

Multiply $2$ by $2$ .

$f (\frac{2 - 5 π}{4}) = 2 \arccos (\frac{2 - 5 π}{4} + \frac{π \cdot 2}{4})$

Step 1.4.2.3

Combine the numerators over the common denominator.

$f (\frac{2 - 5 π}{4}) = 2 \arccos (\frac{2 - 5 π + π \cdot 2}{4})$

Step 1.4.2.4

Simplify the numerator.

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Step 1.4.2.4.1

Move $2$ to the left of $π$ .

$f (\frac{2 - 5 π}{4}) = 2 \arccos (\frac{2 - 5 π + 2 \cdot π}{4})$

Step 1.4.2.4.2

Add $- 5 π$ and $2 π$ .

$f (\frac{2 - 5 π}{4}) = 2 \arccos (\frac{2 - 3 π}{4})$

Step 1.4.2.5

The final answer is $2 \arccos (\frac{2 - 3 π}{4})$ .

$2 \arccos (\frac{2 - 3 π}{4})$

Step 1.5

Find the point at $x = 1 - \frac{3 π}{2}$ .

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Step 1.5.1

Replace the variable $x$ with $1 - \frac{3 π}{2}$ in the expression.

$f (1 - \frac{3 π}{2}) = 2 \arccos ((1 - \frac{3 π}{2}) + \frac{π}{2})$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Combine the numerators over the common denominator.

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 + \frac{- 3 π + π}{2})$

Step 1.5.2.2

Add $- 3 π$ and $π$ .

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 + \frac{- 2 π}{2})$

Step 1.5.2.3

Cancel the common factor of $- 2$ and $2$ .

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Step 1.5.2.3.1

Factor $2$ out of $- 2 π$ .

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 + \frac{2 (- π)}{2})$

Step 1.5.2.3.2

Cancel the common factors.

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Step 1.5.2.3.2.1

Factor $2$ out of $2$ .

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 + \frac{2 (- π)}{2 (1)})$

Step 1.5.2.3.2.2

Cancel the common factor.

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 + \frac{2 (- π)}{2 \cdot 1})$

Step 1.5.2.3.2.3

Rewrite the expression.

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 + \frac{- π}{1})$

Step 1.5.2.3.2.4

Divide $- π$ by $1$ .

$f (1 - \frac{3 π}{2}) = 2 \arccos (1 - π)$

Step 1.5.2.4

The final answer is $2 \arccos (1 - π)$ .

$2 \arccos (1 - π)$

Step 1.6

List the points in a table.

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

Step 2

The trig function can be graphed using the points.

Step 3