Trigonometry Examples

$y = \frac{π}{2} + \arccos (x)$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - 1$ .

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Step 1.1.1

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

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

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

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

$f (- 1) = \frac{π}{2} + π$

Step 1.1.2.2

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

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

Step 1.1.2.3

Combine fractions.

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Step 1.1.2.3.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 1.1.2.3.2

Combine the numerators over the common denominator.

$f (- 1) = \frac{π + π \cdot 2}{2}$

Step 1.1.2.4

Simplify the numerator.

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Step 1.1.2.4.1

Move $2$ to the left of $π$ .

$f (- 1) = \frac{π + 2 \cdot π}{2}$

Step 1.1.2.4.2

Add $π$ and $2 π$ .

$f (- 1) = \frac{3 π}{2}$

Step 1.1.2.5

The final answer is $\frac{3 π}{2}$ .

$\frac{3 π}{2}$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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

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

Step 1.2.2.4

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

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Step 1.2.2.4.1

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

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

Step 1.2.2.4.2

Multiply $2$ by $3$ .

$f (- \frac{1}{2}) = \frac{π \cdot 3}{6} + \frac{2 π}{3} \cdot \frac{2}{2}$

Step 1.2.2.4.3

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

$f (- \frac{1}{2}) = \frac{π \cdot 3}{6} + \frac{2 π \cdot 2}{3 \cdot 2}$

Step 1.2.2.4.4

Multiply $3$ by $2$ .

$f (- \frac{1}{2}) = \frac{π \cdot 3}{6} + \frac{2 π \cdot 2}{6}$

Step 1.2.2.5

Combine the numerators over the common denominator.

$f (- \frac{1}{2}) = \frac{π \cdot 3 + 2 π \cdot 2}{6}$

Step 1.2.2.6

Simplify the numerator.

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Step 1.2.2.6.1

Move $3$ to the left of $π$ .

$f (- \frac{1}{2}) = \frac{3 \cdot π + 2 π \cdot 2}{6}$

Step 1.2.2.6.2

Multiply $2$ by $2$ .

$f (- \frac{1}{2}) = \frac{3 π + 4 π}{6}$

Step 1.2.2.6.3

Add $3 π$ and $4 π$ .

$f (- \frac{1}{2}) = \frac{7 π}{6}$

Step 1.2.2.7

The final answer is $\frac{7 π}{6}$ .

$\frac{7 π}{6}$

Step 1.3

Find the point at $x = 0$ .

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Step 1.3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{π}{2} + \arccos (0)$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

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

$f (0) = \frac{π}{2} + \frac{π}{2}$

Step 1.3.2.2

Combine the numerators over the common denominator.

$f (0) = \frac{π + π}{2}$

Step 1.3.2.3

Add $π$ and $π$ .

$f (0) = \frac{2 π}{2}$

Step 1.3.2.4

Cancel the common factor of $2$ .

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Step 1.3.2.4.1

Cancel the common factor.

$f (0) = \frac{2 π}{2}$

Step 1.3.2.4.2

Divide $π$ by $1$ .

$f (0) = π$

Step 1.3.2.5

The final answer is $π$ .

$π$

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

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

Step 1.4.2.2

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

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

Step 1.4.2.3

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

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

Step 1.4.2.4

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

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Step 1.4.2.4.1

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

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

Step 1.4.2.4.2

Multiply $2$ by $3$ .

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

Step 1.4.2.4.3

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

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

Step 1.4.2.4.4

Multiply $3$ by $2$ .

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

Step 1.4.2.5

Combine the numerators over the common denominator.

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

Step 1.4.2.6

Simplify the numerator.

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Step 1.4.2.6.1

Move $3$ to the left of $π$ .

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

Step 1.4.2.6.2

Move $2$ to the left of $π$ .

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

Step 1.4.2.6.3

Add $3 π$ and $2 π$ .

$f (\frac{1}{2}) = \frac{5 π}{6}$

Step 1.4.2.7

The final answer is $\frac{5 π}{6}$ .

$\frac{5 π}{6}$

Step 1.5

Find the point at $x = 1$ .

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Step 1.5.1

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

$f (1) = \frac{π}{2} + \arccos (1)$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

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

$f (1) = \frac{π}{2} + 0$

Step 1.5.2.2

Add $\frac{π}{2}$ and $0$ .

$f (1) = \frac{π}{2}$

Step 1.5.2.3

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 1 & \frac{3 π}{2} \\ - \frac{1}{2} & \frac{7 π}{6} \\ 0 & π \\ \frac{1}{2} & \frac{5 π}{6} \\ 1 & \frac{π}{2} \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ - 1 & \frac{3 π}{2} \\ - \frac{1}{2} & \frac{7 π}{6} \\ 0 & π \\ \frac{1}{2} & \frac{5 π}{6} \\ 1 & \frac{π}{2} \end{array}$

Step 3