Trigonometry Examples

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

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = 4$ .

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Step 1.1.1

Replace the variable $x$ with $4$ in the expression.

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

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Simplify the numerator.

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Step 1.1.2.1.1

Subtract $5$ from $4$ .

$f (4) = \frac{4 \arcsin (- 1) - π}{4}$

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Cancel the common factor of $2$ .

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Step 1.1.2.1.3.1

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

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

Step 1.1.2.1.3.2

Factor $2$ out of $4$ .

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

Step 1.1.2.1.3.3

Cancel the common factor.

$f (4) = \frac{2 \cdot (2 (\frac{- π}{2})) - π}{4}$

Step 1.1.2.1.3.4

Rewrite the expression.

$f (4) = \frac{2 (- π) - π}{4}$

Step 1.1.2.1.4

Multiply $- 1$ by $2$ .

$f (4) = \frac{- 2 π - π}{4}$

Step 1.1.2.1.5

Subtract $π$ from $- 2 π$ .

$f (4) = \frac{- 3 π}{4}$

Step 1.1.2.2

Move the negative in front of the fraction.

$f (4) = - \frac{3 π}{4}$

Step 1.1.2.3

The final answer is $- \frac{3 π}{4}$ .

$- \frac{3 π}{4}$

Step 1.2

Find the point at $x = \frac{9}{2}$ .

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Step 1.2.1

Replace the variable $x$ with $\frac{9}{2}$ in the expression.

$f (\frac{9}{2}) = \frac{4 \arcsin ((\frac{9}{2}) - 5) - π}{4}$

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 $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{9}{2}) = \frac{4 \arcsin (\frac{9}{2} - 5 \cdot \frac{2}{2}) - π}{4}$

Step 1.2.2.1.2

Combine $- 5$ and $\frac{2}{2}$ .

$f (\frac{9}{2}) = \frac{4 \arcsin (\frac{9}{2} + \frac{- 5 \cdot 2}{2}) - π}{4}$

Step 1.2.2.1.3

Combine the numerators over the common denominator.

$f (\frac{9}{2}) = \frac{4 \arcsin (\frac{9 - 5 \cdot 2}{2}) - π}{4}$

Step 1.2.2.1.4

Simplify the numerator.

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Step 1.2.2.1.4.1

Multiply $- 5$ by $2$ .

$f (\frac{9}{2}) = \frac{4 \arcsin (\frac{9 - 10}{2}) - π}{4}$

Step 1.2.2.1.4.2

Subtract $10$ from $9$ .

$f (\frac{9}{2}) = \frac{4 \arcsin (\frac{- 1}{2}) - π}{4}$

Step 1.2.2.1.5

Move the negative in front of the fraction.

$f (\frac{9}{2}) = \frac{4 \arcsin (- \frac{1}{2}) - π}{4}$

Step 1.2.2.1.6

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

$f (\frac{9}{2}) = \frac{4 (- \frac{π}{6}) - π}{4}$

Step 1.2.2.1.7

Cancel the common factor of $2$ .

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Step 1.2.2.1.7.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$f (\frac{9}{2}) = \frac{4 (\frac{- π}{6}) - π}{4}$

Step 1.2.2.1.7.2

Factor $2$ out of $4$ .

$f (\frac{9}{2}) = \frac{2 (2) (\frac{- π}{6}) - π}{4}$

Step 1.2.2.1.7.3

Factor $2$ out of $6$ .

$f (\frac{9}{2}) = \frac{2 \cdot (2 (\frac{- π}{2 \cdot 3})) - π}{4}$

Step 1.2.2.1.7.4

Cancel the common factor.

$f (\frac{9}{2}) = \frac{2 \cdot (2 (\frac{- π}{2 \cdot 3})) - π}{4}$

Step 1.2.2.1.7.5

Rewrite the expression.

$f (\frac{9}{2}) = \frac{2 (\frac{- π}{3}) - π}{4}$

Step 1.2.2.1.8

Combine $2$ and $\frac{- π}{3}$ .

$f (\frac{9}{2}) = \frac{\frac{2 (- π)}{3} - π}{4}$

Step 1.2.2.1.9

Multiply $- 1$ by $2$ .

$f (\frac{9}{2}) = \frac{\frac{- 2 π}{3} - π}{4}$

Step 1.2.2.1.10

Move the negative in front of the fraction.

$f (\frac{9}{2}) = \frac{- \frac{2 π}{3} - π}{4}$

Step 1.2.2.1.11

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

$f (\frac{9}{2}) = \frac{- \frac{2 π}{3} - π \cdot \frac{3}{3}}{4}$

Step 1.2.2.1.12

Combine $- π$ and $\frac{3}{3}$ .

$f (\frac{9}{2}) = \frac{- \frac{2 π}{3} + \frac{- π \cdot 3}{3}}{4}$

Step 1.2.2.1.13

Combine the numerators over the common denominator.

$f (\frac{9}{2}) = \frac{\frac{- 2 π - π \cdot 3}{3}}{4}$

Step 1.2.2.1.14

Rewrite $\frac{- 2 π - π \cdot 3}{3}$ in a factored form.

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Step 1.2.2.1.14.1

Multiply $3$ by $- 1$ .

$f (\frac{9}{2}) = \frac{\frac{- 2 π - 3 π}{3}}{4}$

Step 1.2.2.1.14.2

Subtract $3 π$ from $- 2 π$ .

$f (\frac{9}{2}) = \frac{\frac{- 5 π}{3}}{4}$

Step 1.2.2.1.15

Move the negative in front of the fraction.

$f (\frac{9}{2}) = \frac{- \frac{5 π}{3}}{4}$

Step 1.2.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9}{2}) = - \frac{5 π}{3} \cdot \frac{1}{4}$

Step 1.2.2.3

Multiply $- \frac{5 π}{3} \cdot \frac{1}{4}$ .

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Step 1.2.2.3.1

Multiply $\frac{1}{4}$ by $\frac{5 π}{3}$ .

$f (\frac{9}{2}) = - \frac{5 π}{4 \cdot 3}$

Step 1.2.2.3.2

Multiply $4$ by $3$ .

$f (\frac{9}{2}) = - \frac{5 π}{12}$

Step 1.2.2.4

The final answer is $- \frac{5 π}{12}$ .

$- \frac{5 π}{12}$

Step 1.3

Find the point at $x = 5$ .

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Step 1.3.1

Replace the variable $x$ with $5$ in the expression.

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

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Simplify the numerator.

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Step 1.3.2.1.1

Subtract $5$ from $5$ .

$f (5) = \frac{4 \arcsin (0) - π}{4}$

Step 1.3.2.1.2

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

$f (5) = \frac{4 \cdot 0 - π}{4}$

Step 1.3.2.1.3

Multiply $4$ by $0$ .

$f (5) = \frac{0 - π}{4}$

Step 1.3.2.1.4

Subtract $π$ from $0$ .

$f (5) = \frac{- π}{4}$

Step 1.3.2.2

Move the negative in front of the fraction.

$f (5) = - \frac{π}{4}$

Step 1.3.2.3

The final answer is $- \frac{π}{4}$ .

$- \frac{π}{4}$

Step 1.4

Find the point at $x = \frac{11}{2}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{11}{2}$ in the expression.

$f (\frac{11}{2}) = \frac{4 \arcsin ((\frac{11}{2}) - 5) - π}{4}$

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 $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{11}{2}) = \frac{4 \arcsin (\frac{11}{2} - 5 \cdot \frac{2}{2}) - π}{4}$

Step 1.4.2.1.2

Combine $- 5$ and $\frac{2}{2}$ .

$f (\frac{11}{2}) = \frac{4 \arcsin (\frac{11}{2} + \frac{- 5 \cdot 2}{2}) - π}{4}$

Step 1.4.2.1.3

Combine the numerators over the common denominator.

$f (\frac{11}{2}) = \frac{4 \arcsin (\frac{11 - 5 \cdot 2}{2}) - π}{4}$

Step 1.4.2.1.4

Simplify the numerator.

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Step 1.4.2.1.4.1

Multiply $- 5$ by $2$ .

$f (\frac{11}{2}) = \frac{4 \arcsin (\frac{11 - 10}{2}) - π}{4}$

Step 1.4.2.1.4.2

Subtract $10$ from $11$ .

$f (\frac{11}{2}) = \frac{4 \arcsin (\frac{1}{2}) - π}{4}$

Step 1.4.2.1.5

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

$f (\frac{11}{2}) = \frac{4 (\frac{π}{6}) - π}{4}$

Step 1.4.2.1.6

Cancel the common factor of $2$ .

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Step 1.4.2.1.6.1

Factor $2$ out of $4$ .

$f (\frac{11}{2}) = \frac{2 (2) (\frac{π}{6}) - π}{4}$

Step 1.4.2.1.6.2

Factor $2$ out of $6$ .

$f (\frac{11}{2}) = \frac{2 \cdot (2 (\frac{π}{2 \cdot 3})) - π}{4}$

Step 1.4.2.1.6.3

Cancel the common factor.

$f (\frac{11}{2}) = \frac{2 \cdot (2 (\frac{π}{2 \cdot 3})) - π}{4}$

Step 1.4.2.1.6.4

Rewrite the expression.

$f (\frac{11}{2}) = \frac{2 (\frac{π}{3}) - π}{4}$

Step 1.4.2.1.7

Combine $2$ and $\frac{π}{3}$ .

$f (\frac{11}{2}) = \frac{\frac{2 π}{3} - π}{4}$

Step 1.4.2.1.8

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

$f (\frac{11}{2}) = \frac{\frac{2 π}{3} - π \cdot \frac{3}{3}}{4}$

Step 1.4.2.1.9

Combine $- π$ and $\frac{3}{3}$ .

$f (\frac{11}{2}) = \frac{\frac{2 π}{3} + \frac{- π \cdot 3}{3}}{4}$

Step 1.4.2.1.10

Combine the numerators over the common denominator.

$f (\frac{11}{2}) = \frac{\frac{2 π - π \cdot 3}{3}}{4}$

Step 1.4.2.1.11

Rewrite $\frac{2 π - π \cdot 3}{3}$ in a factored form.

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Step 1.4.2.1.11.1

Multiply $3$ by $- 1$ .

$f (\frac{11}{2}) = \frac{\frac{2 π - 3 π}{3}}{4}$

Step 1.4.2.1.11.2

Subtract $3 π$ from $2 π$ .

$f (\frac{11}{2}) = \frac{\frac{- π}{3}}{4}$

Step 1.4.2.1.12

Move the negative in front of the fraction.

$f (\frac{11}{2}) = \frac{- \frac{π}{3}}{4}$

Step 1.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{11}{2}) = - \frac{π}{3} \cdot \frac{1}{4}$

Step 1.4.2.3

Multiply $- \frac{π}{3} \cdot \frac{1}{4}$ .

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Step 1.4.2.3.1

Multiply $\frac{1}{4}$ by $\frac{π}{3}$ .

$f (\frac{11}{2}) = - \frac{π}{4 \cdot 3}$

Step 1.4.2.3.2

Multiply $4$ by $3$ .

$f (\frac{11}{2}) = - \frac{π}{12}$

Step 1.4.2.4

The final answer is $- \frac{π}{12}$ .

$- \frac{π}{12}$

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.

$f (6) = \frac{4 \arcsin ((6) - 5) - π}{4}$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Simplify the numerator.

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Step 1.5.2.1.1

Subtract $5$ from $6$ .

$f (6) = \frac{4 \arcsin (1) - π}{4}$

Step 1.5.2.1.2

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

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

Step 1.5.2.1.3

Cancel the common factor of $2$ .

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Step 1.5.2.1.3.1

Factor $2$ out of $4$ .

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

Step 1.5.2.1.3.2

Cancel the common factor.

$f (6) = \frac{2 \cdot (2 (\frac{π}{2})) - π}{4}$

Step 1.5.2.1.3.3

Rewrite the expression.

$f (6) = \frac{2 π - π}{4}$

Step 1.5.2.1.4

Subtract $π$ from $2 π$ .

$f (6) = \frac{π}{4}$

Step 1.5.2.2

The final answer is $\frac{π}{4}$ .

$\frac{π}{4}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 4 & - \frac{3 π}{4} \\ \frac{9}{2} & - \frac{5 π}{12} \\ 5 & - \frac{π}{4} \\ \frac{11}{2} & - \frac{π}{12} \\ 6 & \frac{π}{4} \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ 4 & - \frac{3 π}{4} \\ \frac{9}{2} & - \frac{5 π}{12} \\ 5 & - \frac{π}{4} \\ \frac{11}{2} & - \frac{π}{12} \\ 6 & \frac{π}{4} \end{array}$

Step 3