Trigonometry Examples

$f (x) = \arcsin (\frac{x}{6}) + 3$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - 6$ .

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Step 1.1.1

Replace the variable $x$ with $- 6$ in the expression.

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

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

Divide $- 6$ by $6$ .

$f (- 6) = \arcsin (- 1) + 3$

Step 1.1.2.1.2

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

$f (- 6) = - \frac{π}{2} + 3$

Step 1.1.2.2

The final answer is $- \frac{π}{2} + 3$ .

$- \frac{π}{2} + 3$

Step 1.2

Find the point at $x = - 3$ .

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Step 1.2.1

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

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

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

Cancel the common factor of $- 3$ and $6$ .

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Step 1.2.2.1.1.1

Factor $3$ out of $- 3$ .

$f (- 3) = \arcsin (\frac{3 (- 1)}{6}) + 3$

Step 1.2.2.1.1.2

Cancel the common factors.

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Step 1.2.2.1.1.2.1

Factor $3$ out of $6$ .

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

Step 1.2.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.2.1.1.2.3

Rewrite the expression.

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

Step 1.2.2.1.2

Move the negative in front of the fraction.

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

Step 1.2.2.1.3

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

$f (- 3) = - \frac{π}{6} + 3$

Step 1.2.2.2

The final answer is $- \frac{π}{6} + 3$ .

$- \frac{π}{6} + 3$

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.

$f (0) = \arcsin (\frac{0}{6}) + 3$

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

Divide $0$ by $6$ .

$f (0) = \arcsin (0) + 3$

Step 1.3.2.1.2

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

$f (0) = 0 + 3$

Step 1.3.2.2

Add $0$ and $3$ .

$f (0) = 3$

Step 1.3.2.3

The final answer is $3$ .

$3$

Step 1.4

Find the point at $x = 3$ .

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Step 1.4.1

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

$f (3) = \arcsin (\frac{3}{6}) + 3$

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

Cancel the common factor of $3$ and $6$ .

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Step 1.4.2.1.1.1

Factor $3$ out of $3$ .

$f (3) = \arcsin (\frac{3 (1)}{6}) + 3$

Step 1.4.2.1.1.2

Cancel the common factors.

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Step 1.4.2.1.1.2.1

Factor $3$ out of $6$ .

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

Step 1.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.4.2.1.1.2.3

Rewrite the expression.

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

Step 1.4.2.1.2

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

$f (3) = \frac{π}{6} + 3$

Step 1.4.2.2

The final answer is $\frac{π}{6} + 3$ .

$\frac{π}{6} + 3$

Step 1.5

Find the point at $x = 6$ .

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Step 1.5.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \arcsin (\frac{6}{6}) + 3$

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

Divide $6$ by $6$ .

$f (6) = \arcsin (1) + 3$

Step 1.5.2.1.2

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

$f (6) = \frac{π}{2} + 3$

Step 1.5.2.2

The final answer is $\frac{π}{2} + 3$ .

$\frac{π}{2} + 3$

Step 1.6

List the points in a table.

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

Step 2

The trig function can be graphed using the points.

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

Step 3