Trigonometry Examples

$y = \arcsin (\frac{4}{3} x)$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - \frac{3}{4}$ .

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Step 1.1.1

Replace the variable $x$ with $- \frac{3}{4}$ in the expression.

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

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

Multiply $- 1$ by $4$ .

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

Step 1.1.2.1.2

Combine $- 4$ and $\frac{3}{4}$ .

$f (- \frac{3}{4}) = \arcsin (\frac{\frac{- 4 \cdot 3}{4}}{3})$

Step 1.1.2.2

Simplify the expression.

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Step 1.1.2.2.1

Multiply $- 4$ by $3$ .

$f (- \frac{3}{4}) = \arcsin (\frac{\frac{- 12}{4}}{3})$

Step 1.1.2.2.2

Divide $- 12$ by $4$ .

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

Step 1.1.2.2.3

Divide $- 3$ by $3$ .

$f (- \frac{3}{4}) = \arcsin (- 1)$

Step 1.1.2.3

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

$f (- \frac{3}{4}) = - \frac{π}{2}$

Step 1.1.2.4

The final answer is $- \frac{π}{2}$ .

$- \frac{π}{2}$

Step 1.2

Find the point at $x = - \frac{3}{8}$ .

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Step 1.2.1

Replace the variable $x$ with $- \frac{3}{8}$ in the expression.

$f (- \frac{3}{8}) = \arcsin (\frac{4 (- \frac{3}{8})}{3})$

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

Multiply $- 1$ by $4$ .

$f (- \frac{3}{8}) = \arcsin (\frac{- 4 (\frac{3}{8})}{3})$

Step 1.2.2.1.2

Combine $- 4$ and $\frac{3}{8}$ .

$f (- \frac{3}{8}) = \arcsin (\frac{\frac{- 4 \cdot 3}{8}}{3})$

Step 1.2.2.2

Multiply $- 4$ by $3$ .

$f (- \frac{3}{8}) = \arcsin (\frac{\frac{- 12}{8}}{3})$

Step 1.2.2.3

Reduce the expression by cancelling the common factors.

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Step 1.2.2.3.1

Cancel the common factor of $- 12$ and $8$ .

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Step 1.2.2.3.1.1

Factor $4$ out of $- 12$ .

$f (- \frac{3}{8}) = \arcsin (\frac{\frac{4 (- 3)}{8}}{3})$

Step 1.2.2.3.1.2

Cancel the common factors.

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Step 1.2.2.3.1.2.1

Factor $4$ out of $8$ .

$f (- \frac{3}{8}) = \arcsin (\frac{\frac{4 \cdot - 3}{4 \cdot 2}}{3})$

Step 1.2.2.3.1.2.2

Cancel the common factor.

$f (- \frac{3}{8}) = \arcsin (\frac{\frac{4 \cdot - 3}{4 \cdot 2}}{3})$

Step 1.2.2.3.1.2.3

Rewrite the expression.

$f (- \frac{3}{8}) = \arcsin (\frac{\frac{- 3}{2}}{3})$

Step 1.2.2.3.2

Move the negative in front of the fraction.

$f (- \frac{3}{8}) = \arcsin (\frac{- \frac{3}{2}}{3})$

Step 1.2.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{3}{8}) = \arcsin (- \frac{3}{2} \cdot \frac{1}{3})$

Step 1.2.2.5

Cancel the common factor of $3$ .

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Step 1.2.2.5.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f (- \frac{3}{8}) = \arcsin (\frac{- 3}{2} \cdot \frac{1}{3})$

Step 1.2.2.5.2

Factor $3$ out of $- 3$ .

$f (- \frac{3}{8}) = \arcsin (\frac{3 (- 1)}{2} \cdot \frac{1}{3})$

Step 1.2.2.5.3

Cancel the common factor.

$f (- \frac{3}{8}) = \arcsin (\frac{3 \cdot - 1}{2} \cdot \frac{1}{3})$

Step 1.2.2.5.4

Rewrite the expression.

$f (- \frac{3}{8}) = \arcsin (\frac{- 1}{2})$

Step 1.2.2.6

Move the negative in front of the fraction.

$f (- \frac{3}{8}) = \arcsin (- \frac{1}{2})$

Step 1.2.2.7

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

$f (- \frac{3}{8}) = - \frac{π}{6}$

Step 1.2.2.8

The final answer is $- \frac{π}{6}$ .

$- \frac{π}{6}$

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.

$f (0) = \arcsin (\frac{4 (0)}{3})$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Cancel the common factor of $0$ and $3$ .

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Step 1.3.2.1.1

Factor $3$ out of $4 (0)$ .

$f (0) = \arcsin (\frac{3 (4 \cdot (0))}{3})$

Step 1.3.2.1.2

Cancel the common factors.

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Step 1.3.2.1.2.1

Factor $3$ out of $3$ .

$f (0) = \arcsin (\frac{3 (4 \cdot (0))}{3 (1)})$

Step 1.3.2.1.2.2

Cancel the common factor.

$f (0) = \arcsin (\frac{3 (4 \cdot (0))}{3 \cdot 1})$

Step 1.3.2.1.2.3

Rewrite the expression.

$f (0) = \arcsin (\frac{4 \cdot (0)}{1})$

Step 1.3.2.1.2.4

Divide $4 \cdot (0)$ by $1$ .

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

Step 1.3.2.2

Multiply $4$ by $0$ .

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

Step 1.3.2.3

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

$f (0) = 0$

Step 1.3.2.4

The final answer is $0$ .

$0$

Step 1.4

Find the point at $x = \frac{3}{8}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{3}{8}$ in the expression.

$f (\frac{3}{8}) = \arcsin (\frac{4 (\frac{3}{8})}{3})$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

Combine $4$ and $\frac{3}{8}$ .

$f (\frac{3}{8}) = \arcsin (\frac{\frac{4 \cdot 3}{8}}{3})$

Step 1.4.2.2

Multiply $4$ by $3$ .

$f (\frac{3}{8}) = \arcsin (\frac{\frac{12}{8}}{3})$

Step 1.4.2.3

Cancel the common factor of $12$ and $8$ .

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Step 1.4.2.3.1

Factor $4$ out of $12$ .

$f (\frac{3}{8}) = \arcsin (\frac{\frac{4 (3)}{8}}{3})$

Step 1.4.2.3.2

Cancel the common factors.

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Step 1.4.2.3.2.1

Factor $4$ out of $8$ .

$f (\frac{3}{8}) = \arcsin (\frac{\frac{4 \cdot 3}{4 \cdot 2}}{3})$

Step 1.4.2.3.2.2

Cancel the common factor.

$f (\frac{3}{8}) = \arcsin (\frac{\frac{4 \cdot 3}{4 \cdot 2}}{3})$

Step 1.4.2.3.2.3

Rewrite the expression.

$f (\frac{3}{8}) = \arcsin (\frac{\frac{3}{2}}{3})$

Step 1.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3}{8}) = \arcsin (\frac{3}{2} \cdot \frac{1}{3})$

Step 1.4.2.5

Cancel the common factor of $3$ .

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Step 1.4.2.5.1

Cancel the common factor.

$f (\frac{3}{8}) = \arcsin (\frac{3}{2} \cdot \frac{1}{3})$

Step 1.4.2.5.2

Rewrite the expression.

$f (\frac{3}{8}) = \arcsin (\frac{1}{2})$

Step 1.4.2.6

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

$f (\frac{3}{8}) = \frac{π}{6}$

Step 1.4.2.7

The final answer is $\frac{π}{6}$ .

$\frac{π}{6}$

Step 1.5

Find the point at $x = \frac{3}{4}$ .

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Step 1.5.1

Replace the variable $x$ with $\frac{3}{4}$ in the expression.

$f (\frac{3}{4}) = \arcsin (\frac{4 (\frac{3}{4})}{3})$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Combine $4$ and $\frac{3}{4}$ .

$f (\frac{3}{4}) = \arcsin (\frac{\frac{4 \cdot 3}{4}}{3})$

Step 1.5.2.2

Simplify the expression.

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Step 1.5.2.2.1

Multiply $4$ by $3$ .

$f (\frac{3}{4}) = \arcsin (\frac{\frac{12}{4}}{3})$

Step 1.5.2.2.2

Divide $12$ by $4$ .

$f (\frac{3}{4}) = \arcsin (\frac{3}{3})$

Step 1.5.2.2.3

Divide $3$ by $3$ .

$f (\frac{3}{4}) = \arcsin (1)$

Step 1.5.2.3

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

$f (\frac{3}{4}) = \frac{π}{2}$

Step 1.5.2.4

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{3}{4} & - \frac{π}{2} \\ - \frac{3}{8} & - \frac{π}{6} \\ 0 & 0 \\ \frac{3}{8} & \frac{π}{6} \\ \frac{3}{4} & \frac{π}{2} \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ - \frac{3}{4} & - \frac{π}{2} \\ - \frac{3}{8} & - \frac{π}{6} \\ 0 & 0 \\ \frac{3}{8} & \frac{π}{6} \\ \frac{3}{4} & \frac{π}{2} \end{array}$

Step 3