Trigonometry Examples

$2 \arcsin (x - 4)$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = 3$ .

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Step 1.1.1

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

$f (3) = 2 \arcsin ((3) - 4)$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Subtract $4$ from $3$ .

$f (3) = 2 \arcsin (- 1)$

Step 1.1.2.2

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

$f (3) = 2 (- \frac{π}{2})$

Step 1.1.2.3

Cancel the common factor of $2$ .

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Step 1.1.2.3.1

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

$f (3) = 2 (\frac{- π}{2})$

Step 1.1.2.3.2

Cancel the common factor.

$f (3) = 2 (\frac{- π}{2})$

Step 1.1.2.3.3

Rewrite the expression.

$f (3) = - π$

Step 1.1.2.4

The final answer is $- π$ .

$- π$

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.

$f (\frac{7}{2}) = 2 \arcsin ((\frac{7}{2}) - 4)$

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{7}{2}) = 2 \arcsin (\frac{7}{2} - 4 \cdot \frac{2}{2})$

Step 1.2.2.2

Combine $- 4$ and $\frac{2}{2}$ .

$f (\frac{7}{2}) = 2 \arcsin (\frac{7}{2} + \frac{- 4 \cdot 2}{2})$

Step 1.2.2.3

Combine the numerators over the common denominator.

$f (\frac{7}{2}) = 2 \arcsin (\frac{7 - 4 \cdot 2}{2})$

Step 1.2.2.4

Simplify the numerator.

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Step 1.2.2.4.1

Multiply $- 4$ by $2$ .

$f (\frac{7}{2}) = 2 \arcsin (\frac{7 - 8}{2})$

Step 1.2.2.4.2

Subtract $8$ from $7$ .

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

Step 1.2.2.5

Move the negative in front of the fraction.

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

Step 1.2.2.6

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

$f (\frac{7}{2}) = 2 (- \frac{π}{6})$

Step 1.2.2.7

Cancel the common factor of $2$ .

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Step 1.2.2.7.1

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

$f (\frac{7}{2}) = 2 (\frac{- π}{6})$

Step 1.2.2.7.2

Factor $2$ out of $6$ .

$f (\frac{7}{2}) = 2 (\frac{- π}{2 (3)})$

Step 1.2.2.7.3

Cancel the common factor.

$f (\frac{7}{2}) = 2 (\frac{- π}{2 \cdot 3})$

Step 1.2.2.7.4

Rewrite the expression.

$f (\frac{7}{2}) = \frac{- π}{3}$

Step 1.2.2.8

Move the negative in front of the fraction.

$f (\frac{7}{2}) = - \frac{π}{3}$

Step 1.2.2.9

The final answer is $- \frac{π}{3}$ .

$- \frac{π}{3}$

Step 1.3

Find the point at $x = 4$ .

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Step 1.3.1

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

$f (4) = 2 \arcsin ((4) - 4)$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Subtract $4$ from $4$ .

$f (4) = 2 \arcsin (0)$

Step 1.3.2.2

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

$f (4) = 2 \cdot 0$

Step 1.3.2.3

Multiply $2$ by $0$ .

$f (4) = 0$

Step 1.3.2.4

The final answer is $0$ .

$0$

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.4.2.2

Combine $- 4$ and $\frac{2}{2}$ .

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

Step 1.4.2.3

Combine the numerators over the common denominator.

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

Step 1.4.2.4

Simplify the numerator.

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Step 1.4.2.4.1

Multiply $- 4$ by $2$ .

$f (\frac{9}{2}) = 2 \arcsin (\frac{9 - 8}{2})$

Step 1.4.2.4.2

Subtract $8$ from $9$ .

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

Step 1.4.2.5

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

$f (\frac{9}{2}) = 2 (\frac{π}{6})$

Step 1.4.2.6

Cancel the common factor of $2$ .

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Step 1.4.2.6.1

Factor $2$ out of $6$ .

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

Step 1.4.2.6.2

Cancel the common factor.

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

Step 1.4.2.6.3

Rewrite the expression.

$f (\frac{9}{2}) = \frac{π}{3}$

Step 1.4.2.7

The final answer is $\frac{π}{3}$ .

$\frac{π}{3}$

Step 1.5

Find the point at $x = 5$ .

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Step 1.5.1

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

$f (5) = 2 \arcsin ((5) - 4)$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Subtract $4$ from $5$ .

$f (5) = 2 \arcsin (1)$

Step 1.5.2.2

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

$f (5) = 2 (\frac{π}{2})$

Step 1.5.2.3

Cancel the common factor of $2$ .

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Step 1.5.2.3.1

Cancel the common factor.

$f (5) = 2 (\frac{π}{2})$

Step 1.5.2.3.2

Rewrite the expression.

$f (5) = π$

Step 1.5.2.4

The final answer is $π$ .

$π$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 3 & - π \\ \frac{7}{2} & - \frac{π}{3} \\ 4 & 0 \\ \frac{9}{2} & \frac{π}{3} \\ 5 & π \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ 3 & - π \\ \frac{7}{2} & - \frac{π}{3} \\ 4 & 0 \\ \frac{9}{2} & \frac{π}{3} \\ 5 & π \end{array}$

Step 3