Trigonometry Examples

$3 \arcsin (6 x + π)$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - \frac{1}{6} - \frac{π}{6}$ .

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Step 1.1.1

Replace the variable $x$ with $- \frac{1}{6} - \frac{π}{6}$ in the expression.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (6 (- \frac{1}{6} - \frac{π}{6}) + π)$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Apply the distributive property.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (6 (- \frac{1}{6}) + 6 (- \frac{π}{6}) + π)$

Step 1.1.2.1.2

Cancel the common factor of $6$ .

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Step 1.1.2.1.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (6 (\frac{- 1}{6}) + 6 (- \frac{π}{6}) + π)$

Step 1.1.2.1.2.2

Cancel the common factor.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (6 (\frac{- 1}{6}) + 6 (- \frac{π}{6}) + π)$

Step 1.1.2.1.2.3

Rewrite the expression.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (- 1 + 6 (- \frac{π}{6}) + π)$

Step 1.1.2.1.3

Cancel the common factor of $6$ .

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Step 1.1.2.1.3.1

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

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (- 1 + 6 (\frac{- π}{6}) + π)$

Step 1.1.2.1.3.2

Cancel the common factor.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (- 1 + 6 (\frac{- π}{6}) + π)$

Step 1.1.2.1.3.3

Rewrite the expression.

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (- 1 - π + π)$

Step 1.1.2.2

Simplify by adding terms.

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Step 1.1.2.2.1

Add $- π$ and $π$ .

$f (- \frac{1}{6} - \frac{π}{6}) = 3 \arcsin (- 1 + 0)$

Step 1.1.2.2.2

Add $- 1$ and $0$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Multiply $3 (- \frac{π}{2})$ .

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Step 1.1.2.4.1

Multiply $- 1$ by $3$ .

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

Step 1.1.2.4.2

Combine $- 3$ and $\frac{π}{2}$ .

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

Step 1.1.2.5

Move the negative in front of the fraction.

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

Step 1.1.2.6

The final answer is $- \frac{3 π}{2}$ .

$- \frac{3 π}{2}$

Step 1.2

Find the point at $x = - \frac{1 + 3 π}{12}$ .

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Step 1.2.1

Replace the variable $x$ with $- \frac{1 + 3 π}{12}$ in the expression.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (6 (- \frac{1 + 3 π}{12}) + π)$

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1.1

Move the leading negative in $- \frac{1 + 3 π}{12}$ into the numerator.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (6 (\frac{- (1 + 3 π)}{12}) + π)$

Step 1.2.2.1.1.2

Factor $6$ out of $12$ .

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (6 (\frac{- (1 + 3 π)}{6 (2)}) + π)$

Step 1.2.2.1.1.3

Cancel the common factor.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (6 (\frac{- (1 + 3 π)}{6 \cdot 2}) + π)$

Step 1.2.2.1.1.4

Rewrite the expression.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- (1 + 3 π)}{2} + π)$

Step 1.2.2.1.2

Move the negative in front of the fraction.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (- \frac{1 + 3 π}{2} + π)$

Step 1.2.2.2

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

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (- \frac{1 + 3 π}{2} + π \cdot \frac{2}{2})$

Step 1.2.2.3

Combine fractions.

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Step 1.2.2.3.1

Combine $π$ and $\frac{2}{2}$ .

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (- \frac{1 + 3 π}{2} + \frac{π \cdot 2}{2})$

Step 1.2.2.3.2

Combine the numerators over the common denominator.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- (1 + 3 π) + π \cdot 2}{2})$

Step 1.2.2.4

Simplify the numerator.

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Step 1.2.2.4.1

Apply the distributive property.

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- 1 \cdot 1 - (3 π) + π \cdot 2}{2})$

Step 1.2.2.4.2

Multiply $- 1$ by $1$ .

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- 1 - (3 π) + π \cdot 2}{2})$

Step 1.2.2.4.3

Multiply $3$ by $- 1$ .

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- 1 - 3 π + π \cdot 2}{2})$

Step 1.2.2.4.4

Move $2$ to the left of $π$ .

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- 1 - 3 π + 2 \cdot π}{2})$

Step 1.2.2.4.5

Add $- 3 π$ and $2 π$ .

$f (- \frac{1 + 3 π}{12}) = 3 \arcsin (\frac{- 1 - π}{2})$

Step 1.2.2.5

The final answer is $3 \arcsin (\frac{- 1 - π}{2})$ .

$3 \arcsin (\frac{- 1 - π}{2})$

Step 1.3

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

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Step 1.3.1

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

$f (- \frac{π}{3}) = 3 \arcsin (6 (- \frac{π}{3}) + π)$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1.1

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

$f (- \frac{π}{3}) = 3 \arcsin (6 (\frac{- π}{3}) + π)$

Step 1.3.2.1.1.2

Factor $3$ out of $6$ .

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

Step 1.3.2.1.1.3

Cancel the common factor.

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

Step 1.3.2.1.1.4

Rewrite the expression.

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

Step 1.3.2.1.2

Multiply $- 1$ by $2$ .

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

Step 1.3.2.2

Add $- 2 π$ and $π$ .

$f (- \frac{π}{3}) = 3 \arcsin (- π)$

Step 1.3.2.3

The final answer is $3 \arcsin (- π)$ .

$3 \arcsin (- π)$

Step 1.4

Find the point at $x = \frac{1 - 5 π}{12}$ .

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Step 1.4.1

Replace the variable $x$ with $\frac{1 - 5 π}{12}$ in the expression.

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (6 (\frac{1 - 5 π}{12}) + π)$

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

Cancel the common factor of $6$ .

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Step 1.4.2.1.1

Factor $6$ out of $12$ .

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (6 (\frac{1 - 5 π}{6 (2)}) + π)$

Step 1.4.2.1.2

Cancel the common factor.

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (6 (\frac{1 - 5 π}{6 \cdot 2}) + π)$

Step 1.4.2.1.3

Rewrite the expression.

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (\frac{1 - 5 π}{2} + π)$

Step 1.4.2.2

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

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (\frac{1 - 5 π}{2} + π \cdot \frac{2}{2})$

Step 1.4.2.3

Combine fractions.

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Step 1.4.2.3.1

Combine $π$ and $\frac{2}{2}$ .

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (\frac{1 - 5 π}{2} + \frac{π \cdot 2}{2})$

Step 1.4.2.3.2

Combine the numerators over the common denominator.

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (\frac{1 - 5 π + π \cdot 2}{2})$

Step 1.4.2.4

Simplify the numerator.

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Step 1.4.2.4.1

Move $2$ to the left of $π$ .

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (\frac{1 - 5 π + 2 \cdot π}{2})$

Step 1.4.2.4.2

Add $- 5 π$ and $2 π$ .

$f (\frac{1 - 5 π}{12}) = 3 \arcsin (\frac{1 - 3 π}{2})$

Step 1.4.2.5

The final answer is $3 \arcsin (\frac{1 - 3 π}{2})$ .

$3 \arcsin (\frac{1 - 3 π}{2})$

Step 1.5

Find the point at $x = \frac{1 - 3 π}{6}$ .

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Step 1.5.1

Replace the variable $x$ with $\frac{1 - 3 π}{6}$ in the expression.

$f (\frac{1 - 3 π}{6}) = 3 \arcsin (6 (\frac{1 - 3 π}{6}) + π)$

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

Cancel the common factor of $6$ .

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Step 1.5.2.1.1

Cancel the common factor.

$f (\frac{1 - 3 π}{6}) = 3 \arcsin (6 (\frac{1 - 3 π}{6}) + π)$

Step 1.5.2.1.2

Rewrite the expression.

$f (\frac{1 - 3 π}{6}) = 3 \arcsin (1 - 3 π + π)$

Step 1.5.2.2

Add $- 3 π$ and $π$ .

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

Step 1.5.2.3

The final answer is $3 \arcsin (1 - 2 π)$ .

$3 \arcsin (1 - 2 π)$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{1 - 3 π}{6} & 3 \arcsin (1 - 2 π) \\ \frac{1 - 5 π}{12} & 3 \arcsin (\frac{1 - 3 π}{2}) \\ - \frac{π}{3} & 3 \arcsin (- π) \\ - \frac{1 + 3 π}{12} & 3 \arcsin (\frac{- 1 - π}{2}) \\ - \frac{1}{6} - \frac{π}{6} & - \frac{3 π}{2} \end{array}$

Step 2

The trig function can be graphed using the points.

Step 3