Trigonometry Examples

$y = \arcsin (- x - 2) + 2 π$

Select a few points to graph.

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Find the point at $x = - 3$ .

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Replace the variable $x$ with $- 3$ in the expression.

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

Simplify the result.

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Simplify each term.

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Multiply $- 1$ by $- 3$ .

$f (- 3) = \arcsin (3 - 2) + 2 π$

Subtract $2$ from $3$ .

$f (- 3) = \arcsin (1) + 2 π$

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

$f (- 3) = \frac{π}{2} + 2 π$

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

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

Combine fractions.

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Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$f (- 3) = \frac{π + 2 π \cdot 2}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$f (- 3) = \frac{π + 4 π}{2}$

Add $π$ and $4 π$ .

$f (- 3) = \frac{5 π}{2}$

The final answer is $\frac{5 π}{2}$ .

$\frac{5 π}{2}$

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

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

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

Simplify the result.

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Simplify each term.

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Multiply $- (- \frac{5}{2})$ .

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Multiply $- 1$ by $- 1$ .

$f (- \frac{5}{2}) = \arcsin (1 (\frac{5}{2}) - 2) + 2 π$

Multiply $\frac{5}{2}$ by $1$ .

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

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

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

Combine $- 2$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 2$ by $2$ .

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

Subtract $4$ from $5$ .

$f (- \frac{5}{2}) = \arcsin (\frac{1}{2}) + 2 π$

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

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

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Combine fractions.

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Combine $2 π$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $6$ by $2$ .

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

Add $π$ and $12 π$ .

$f (- \frac{5}{2}) = \frac{13 π}{6}$

The final answer is $\frac{13 π}{6}$ .

$\frac{13 π}{6}$

Find the point at $x = - 2$ .

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

$f (- 2) = \arcsin (- (- 2) - 2) + 2 π$

Simplify the result.

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Simplify each term.

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Multiply $- 1$ by $- 2$ .

$f (- 2) = \arcsin (2 - 2) + 2 π$

Subtract $2$ from $2$ .

$f (- 2) = \arcsin (0) + 2 π$

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

$f (- 2) = 0 + 2 π$

Add $0$ and $2 π$ .

$f (- 2) = 2 π$

The final answer is $2 π$ .

$2 π$

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

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

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

Simplify the result.

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Simplify each term.

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Multiply $- (- \frac{3}{2})$ .

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Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{3}{2}$ by $1$ .

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

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

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

Combine $- 2$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 2$ by $2$ .

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

Subtract $4$ from $3$ .

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

Move the negative in front of the fraction.

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

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

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

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Combine fractions.

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Combine $2 π$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $6$ by $2$ .

$f (- \frac{3}{2}) = \frac{- π + 12 π}{6}$

Add $- π$ and $12 π$ .

$f (- \frac{3}{2}) = \frac{11 π}{6}$

The final answer is $\frac{11 π}{6}$ .

$\frac{11 π}{6}$

Find the point at $x = - 1$ .

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

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

Simplify the result.

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Simplify each term.

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Multiply $- 1$ by $- 1$ .

$f (- 1) = \arcsin (1 - 2) + 2 π$

Subtract $2$ from $1$ .

$f (- 1) = \arcsin (- 1) + 2 π$

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

$f (- 1) = - \frac{π}{2} + 2 π$

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

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

Combine fractions.

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Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $2$ by $2$ .

$f (- 1) = \frac{- π + 4 π}{2}$

Add $- π$ and $4 π$ .

$f (- 1) = \frac{3 π}{2}$

The final answer is $\frac{3 π}{2}$ .

$\frac{3 π}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 3 & \frac{5 π}{2} \\ - \frac{5}{2} & \frac{13 π}{6} \\ - 2 & 2 π \\ - \frac{3}{2} & \frac{11 π}{6} \\ - 1 & \frac{3 π}{2} \end{array}$

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ - 3 & \frac{5 π}{2} \\ - \frac{5}{2} & \frac{13 π}{6} \\ - 2 & 2 π \\ - \frac{3}{2} & \frac{11 π}{6} \\ - 1 & \frac{3 π}{2} \end{array}$