Trigonometry Examples

$y = \arcsin (x + 1) + π$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - 2$ .

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Step 1.1.1

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

$f (- 2) = \arcsin ((- 2) + 1) + π$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Add $- 2$ and $1$ .

$f (- 2) = \arcsin (- 1) + π$

Step 1.1.2.1.2

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

$f (- 2) = - \frac{π}{2} + π$

Step 1.1.2.2

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

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

Step 1.1.2.3

Combine fractions.

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Step 1.1.2.3.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 1.1.2.3.2

Combine the numerators over the common denominator.

$f (- 2) = \frac{- π + π \cdot 2}{2}$

Step 1.1.2.4

Simplify the numerator.

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Step 1.1.2.4.1

Move $2$ to the left of $π$ .

$f (- 2) = \frac{- π + 2 \cdot π}{2}$

Step 1.1.2.4.2

Add $- π$ and $2 π$ .

$f (- 2) = \frac{π}{2}$

Step 1.1.2.5

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2.1.2

Combine the numerators over the common denominator.

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

Step 1.2.2.1.3

Add $- 3$ and $2$ .

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

Step 1.2.2.1.4

Move the negative in front of the fraction.

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

Step 1.2.2.1.5

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

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

Step 1.2.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 1.2.2.3

Combine fractions.

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Step 1.2.2.3.1

Combine $π$ and $\frac{6}{6}$ .

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

Step 1.2.2.3.2

Combine the numerators over the common denominator.

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

Step 1.2.2.4

Simplify the numerator.

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Step 1.2.2.4.1

Move $6$ to the left of $π$ .

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

Step 1.2.2.4.2

Add $- π$ and $6 π$ .

$f (- \frac{3}{2}) = \frac{5 π}{6}$

Step 1.2.2.5

The final answer is $\frac{5 π}{6}$ .

$\frac{5 π}{6}$

Step 1.3

Find the point at $x = - 1$ .

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Step 1.3.1

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

$f (- 1) = \arcsin ((- 1) + 1) + π$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Add $- 1$ and $1$ .

$f (- 1) = \arcsin (0) + π$

Step 1.3.2.1.2

The exact value of $\arcsin (0)$ is $0$ .

$f (- 1) = 0 + π$

Step 1.3.2.2

Add $0$ and $π$ .

$f (- 1) = π$

Step 1.3.2.3

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.

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

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.4.2.1.2

Combine the numerators over the common denominator.

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

Step 1.4.2.1.3

Add $- 1$ and $2$ .

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

Step 1.4.2.1.4

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$f (- \frac{1}{2}) = \frac{π}{6} + π$

Step 1.4.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 1.4.2.3

Combine fractions.

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Step 1.4.2.3.1

Combine $π$ and $\frac{6}{6}$ .

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

Step 1.4.2.3.2

Combine the numerators over the common denominator.

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

Step 1.4.2.4

Simplify the numerator.

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Step 1.4.2.4.1

Move $6$ to the left of $π$ .

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

Step 1.4.2.4.2

Add $π$ and $6 π$ .

$f (- \frac{1}{2}) = \frac{7 π}{6}$

Step 1.4.2.5

The final answer is $\frac{7 π}{6}$ .

$\frac{7 π}{6}$

Step 1.5

Find the point at $x = 0$ .

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Step 1.5.1

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

$f (0) = \arcsin ((0) + 1) + π$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

Add $0$ and $1$ .

$f (0) = \arcsin (1) + π$

Step 1.5.2.1.2

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

$f (0) = \frac{π}{2} + π$

Step 1.5.2.2

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

$f (0) = \frac{π}{2} + π \cdot \frac{2}{2}$

Step 1.5.2.3

Combine fractions.

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Step 1.5.2.3.1

Combine $π$ and $\frac{2}{2}$ .

$f (0) = \frac{π}{2} + \frac{π \cdot 2}{2}$

Step 1.5.2.3.2

Combine the numerators over the common denominator.

$f (0) = \frac{π + π \cdot 2}{2}$

Step 1.5.2.4

Simplify the numerator.

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Step 1.5.2.4.1

Move $2$ to the left of $π$ .

$f (0) = \frac{π + 2 \cdot π}{2}$

Step 1.5.2.4.2

Add $π$ and $2 π$ .

$f (0) = \frac{3 π}{2}$

Step 1.5.2.5

The final answer is $\frac{3 π}{2}$ .

$\frac{3 π}{2}$

Step 1.6

List the points in a table.

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

Step 2

The trig function can be graphed using the points.

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

Step 3