Trigonometry Examples

$y = \arcsin (\frac{x - 6}{5})$

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.

$f (1) = \arcsin (\frac{(1) - 6}{5})$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Subtract $6$ from $1$ .

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

Step 1.1.2.2

Divide $- 5$ by $5$ .

$f (1) = \arcsin (- 1)$

Step 1.1.2.3

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

$f (1) = - \frac{π}{2}$

Step 1.1.2.4

The final answer is $- \frac{π}{2}$ .

$- \frac{π}{2}$

Step 1.2

Find the point at $x = \frac{7}{2}$ .

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Step 1.2.1

Replace the variable $x$ with $\frac{7}{2}$ in the expression.

$f (\frac{7}{2}) = \arcsin (\frac{(\frac{7}{2}) - 6}{5})$

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

Simplify the numerator.

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Step 1.2.2.1.1

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

$f (\frac{7}{2}) = \arcsin (\frac{\frac{7}{2} - 6 \cdot \frac{2}{2}}{5})$

Step 1.2.2.1.2

Combine $- 6$ and $\frac{2}{2}$ .

$f (\frac{7}{2}) = \arcsin (\frac{\frac{7}{2} + \frac{- 6 \cdot 2}{2}}{5})$

Step 1.2.2.1.3

Combine the numerators over the common denominator.

$f (\frac{7}{2}) = \arcsin (\frac{\frac{7 - 6 \cdot 2}{2}}{5})$

Step 1.2.2.1.4

Simplify the numerator.

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Step 1.2.2.1.4.1

Multiply $- 6$ by $2$ .

$f (\frac{7}{2}) = \arcsin (\frac{\frac{7 - 12}{2}}{5})$

Step 1.2.2.1.4.2

Subtract $12$ from $7$ .

$f (\frac{7}{2}) = \arcsin (\frac{\frac{- 5}{2}}{5})$

Step 1.2.2.1.5

Move the negative in front of the fraction.

$f (\frac{7}{2}) = \arcsin (\frac{- \frac{5}{2}}{5})$

Step 1.2.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7}{2}) = \arcsin (- \frac{5}{2} \cdot \frac{1}{5})$

Step 1.2.2.3

Cancel the common factor of $5$ .

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Step 1.2.2.3.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f (\frac{7}{2}) = \arcsin (\frac{- 5}{2} \cdot \frac{1}{5})$

Step 1.2.2.3.2

Factor $5$ out of $- 5$ .

$f (\frac{7}{2}) = \arcsin (\frac{5 (- 1)}{2} \cdot \frac{1}{5})$

Step 1.2.2.3.3

Cancel the common factor.

$f (\frac{7}{2}) = \arcsin (\frac{5 \cdot - 1}{2} \cdot \frac{1}{5})$

Step 1.2.2.3.4

Rewrite the expression.

$f (\frac{7}{2}) = \arcsin (\frac{- 1}{2})$

Step 1.2.2.4

Move the negative in front of the fraction.

$f (\frac{7}{2}) = \arcsin (- \frac{1}{2})$

Step 1.2.2.5

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

$f (\frac{7}{2}) = - \frac{π}{6}$

Step 1.2.2.6

The final answer is $- \frac{π}{6}$ .

$- \frac{π}{6}$

Step 1.3

Find the point at $x = 6$ .

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Step 1.3.1

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

$f (6) = \arcsin (\frac{(6) - 6}{5})$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Subtract $6$ from $6$ .

$f (6) = \arcsin (\frac{0}{5})$

Step 1.3.2.2

Divide $0$ by $5$ .

$f (6) = \arcsin (0)$

Step 1.3.2.3

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

$f (6) = 0$

Step 1.3.2.4

The final answer is $0$ .

$0$

Step 1.4

Find the point at $x = \frac{17}{2}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{17}{2}$ in the expression.

$f (\frac{17}{2}) = \arcsin (\frac{(\frac{17}{2}) - 6}{5})$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

Simplify the numerator.

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Step 1.4.2.1.1

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

$f (\frac{17}{2}) = \arcsin (\frac{\frac{17}{2} - 6 \cdot \frac{2}{2}}{5})$

Step 1.4.2.1.2

Combine $- 6$ and $\frac{2}{2}$ .

$f (\frac{17}{2}) = \arcsin (\frac{\frac{17}{2} + \frac{- 6 \cdot 2}{2}}{5})$

Step 1.4.2.1.3

Combine the numerators over the common denominator.

$f (\frac{17}{2}) = \arcsin (\frac{\frac{17 - 6 \cdot 2}{2}}{5})$

Step 1.4.2.1.4

Simplify the numerator.

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Step 1.4.2.1.4.1

Multiply $- 6$ by $2$ .

$f (\frac{17}{2}) = \arcsin (\frac{\frac{17 - 12}{2}}{5})$

Step 1.4.2.1.4.2

Subtract $12$ from $17$ .

$f (\frac{17}{2}) = \arcsin (\frac{\frac{5}{2}}{5})$

Step 1.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{17}{2}) = \arcsin (\frac{5}{2} \cdot \frac{1}{5})$

Step 1.4.2.3

Cancel the common factor of $5$ .

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Step 1.4.2.3.1

Cancel the common factor.

$f (\frac{17}{2}) = \arcsin (\frac{5}{2} \cdot \frac{1}{5})$

Step 1.4.2.3.2

Rewrite the expression.

$f (\frac{17}{2}) = \arcsin (\frac{1}{2})$

Step 1.4.2.4

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

$f (\frac{17}{2}) = \frac{π}{6}$

Step 1.4.2.5

The final answer is $\frac{π}{6}$ .

$\frac{π}{6}$

Step 1.5

Find the point at $x = 11$ .

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Step 1.5.1

Replace the variable $x$ with $11$ in the expression.

$f (11) = \arcsin (\frac{(11) - 6}{5})$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Subtract $6$ from $11$ .

$f (11) = \arcsin (\frac{5}{5})$

Step 1.5.2.2

Divide $5$ by $5$ .

$f (11) = \arcsin (1)$

Step 1.5.2.3

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

$f (11) = \frac{π}{2}$

Step 1.5.2.4

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 1 & - \frac{π}{2} \\ \frac{7}{2} & - \frac{π}{6} \\ 6 & 0 \\ \frac{17}{2} & \frac{π}{6} \\ 11 & \frac{π}{2} \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ 1 & - \frac{π}{2} \\ \frac{7}{2} & - \frac{π}{6} \\ 6 & 0 \\ \frac{17}{2} & \frac{π}{6} \\ 11 & \frac{π}{2} \end{array}$

Step 3