Trigonometry Examples

$y = \frac{3}{5} \cdot \arcsin (x)$

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.

$f (- 1) = \frac{3 \arcsin (- 1)}{5}$

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

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

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

Step 1.1.2.1.2

Combine exponents.

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Step 1.1.2.1.2.1

Factor out negative.

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

Step 1.1.2.1.2.2

Combine $3$ and $\frac{π}{2}$ .

$f (- 1) = \frac{- \frac{3 π}{2}}{5}$

Step 1.1.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.2.3

Multiply $- \frac{3 π}{2} \cdot \frac{1}{5}$ .

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Step 1.1.2.3.1

Multiply $\frac{1}{5}$ by $\frac{3 π}{2}$ .

$f (- 1) = - \frac{3 π}{5 \cdot 2}$

Step 1.1.2.3.2

Multiply $5$ by $2$ .

$f (- 1) = - \frac{3 π}{10}$

Step 1.1.2.4

The final answer is $- \frac{3 π}{10}$ .

$- \frac{3 π}{10}$

Step 1.2

Find the point at $x = - \frac{1}{2}$ .

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Step 1.2.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$f (- \frac{1}{2}) = \frac{3 \arcsin (- \frac{1}{2})}{5}$

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

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

$f (- \frac{1}{2}) = \frac{3 \cdot (- 1 \frac{π}{6})}{5}$

Step 1.2.2.1.2

Combine exponents.

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Step 1.2.2.1.2.1

Factor out negative.

$f (- \frac{1}{2}) = \frac{- (3 (\frac{π}{6}))}{5}$

Step 1.2.2.1.2.2

Combine $3$ and $\frac{π}{6}$ .

$f (- \frac{1}{2}) = \frac{- \frac{3 π}{6}}{5}$

Step 1.2.2.1.3

Reduce the expression $\frac{3 π}{6}$ by cancelling the common factors.

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Step 1.2.2.1.3.1

Factor $3$ out of $3 π$ .

$f (- \frac{1}{2}) = \frac{- \frac{3 (π)}{6}}{5}$

Step 1.2.2.1.3.2

Factor $3$ out of $6$ .

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

Step 1.2.2.1.3.3

Cancel the common factor.

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

Step 1.2.2.1.3.4

Rewrite the expression.

$f (- \frac{1}{2}) = \frac{- \frac{π}{2}}{5}$

Step 1.2.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.2.3

Multiply $- \frac{π}{2} \cdot \frac{1}{5}$ .

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Step 1.2.2.3.1

Multiply $\frac{1}{5}$ by $\frac{π}{2}$ .

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

Step 1.2.2.3.2

Multiply $5$ by $2$ .

$f (- \frac{1}{2}) = - \frac{π}{10}$

Step 1.2.2.4

The final answer is $- \frac{π}{10}$ .

$- \frac{π}{10}$

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.

$f (0) = \frac{3 \arcsin (0)}{5}$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

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

$f (0) = \frac{3 \cdot 0}{5}$

Step 1.3.2.2

Multiply $3$ by $0$ .

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

Step 1.3.2.3

Divide $0$ by $5$ .

$f (0) = 0$

Step 1.3.2.4

The final answer is $0$ .

$0$

Step 1.4

Find the point at $x = \frac{1}{2}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = \frac{3 \arcsin (\frac{1}{2})}{5}$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

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

$f (\frac{1}{2}) = \frac{3 (\frac{π}{6})}{5}$

Step 1.4.2.2

Combine $3$ and $\frac{π}{6}$ .

$f (\frac{1}{2}) = \frac{\frac{3 π}{6}}{5}$

Step 1.4.2.3

Reduce the expression $\frac{3 π}{6}$ by cancelling the common factors.

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Step 1.4.2.3.1

Factor $3$ out of $3 π$ .

$f (\frac{1}{2}) = \frac{\frac{3 (π)}{6}}{5}$

Step 1.4.2.3.2

Factor $3$ out of $6$ .

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

Step 1.4.2.3.3

Cancel the common factor.

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

Step 1.4.2.3.4

Rewrite the expression.

$f (\frac{1}{2}) = \frac{\frac{π}{2}}{5}$

Step 1.4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4.2.5

Multiply $\frac{π}{2} \cdot \frac{1}{5}$ .

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Step 1.4.2.5.1

Multiply $\frac{π}{2}$ by $\frac{1}{5}$ .

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

Step 1.4.2.5.2

Multiply $2$ by $5$ .

$f (\frac{1}{2}) = \frac{π}{10}$

Step 1.4.2.6

The final answer is $\frac{π}{10}$ .

$\frac{π}{10}$

Step 1.5

Find the point at $x = 1$ .

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Step 1.5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{3 \arcsin (1)}{5}$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

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

$f (1) = \frac{3 (\frac{π}{2})}{5}$

Step 1.5.2.2

Combine $3$ and $\frac{π}{2}$ .

$f (1) = \frac{\frac{3 π}{2}}{5}$

Step 1.5.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.2.4

Multiply $\frac{3 π}{2} \cdot \frac{1}{5}$ .

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Step 1.5.2.4.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{5}$ .

$f (1) = \frac{3 π}{2 \cdot 5}$

Step 1.5.2.4.2

Multiply $2$ by $5$ .

$f (1) = \frac{3 π}{10}$

Step 1.5.2.5

The final answer is $\frac{3 π}{10}$ .

$\frac{3 π}{10}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 1 & - \frac{3 π}{10} \\ - \frac{1}{2} & - \frac{π}{10} \\ 0 & 0 \\ \frac{1}{2} & \frac{π}{10} \\ 1 & \frac{3 π}{10} \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ - 1 & - \frac{3 π}{10} \\ - \frac{1}{2} & - \frac{π}{10} \\ 0 & 0 \\ \frac{1}{2} & \frac{π}{10} \\ 1 & \frac{3 π}{10} \end{array}$

Step 3