Linear Algebra Examples

$x = - 3 \sin (y)$

Step 1

Rewrite the equation as $- 3 \sin (y) = x$ .

$- 3 \sin (y) = x$

Step 2

Divide each term in $- 3 \sin (y) = x$ by $- 3$ and simplify.

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Step 2.1

Divide each term in $- 3 \sin (y) = x$ by $- 3$ .

$\frac{- 3 \sin (y)}{- 3} = \frac{x}{- 3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 3 \sin (y)}{- 3} = \frac{x}{- 3}$

Step 2.2.1.2

Divide $\sin (y)$ by $1$ .

$\sin (y) = \frac{x}{- 3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\sin (y) = - \frac{x}{3}$

Step 3

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

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

Step 4

Set the argument in $\arcsin (- \frac{x}{3})$ greater than or equal to $- 1$ to find where the expression is defined.

$- \frac{x}{3} \geq - 1$

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $- \frac{x}{3} \geq - 1$ by $- 1$ and simplify.

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Step 5.1.1

Divide each term in $- \frac{x}{3} \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{3}}{- 1} \leq \frac{- 1}{- 1}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{3}}{1} \leq \frac{- 1}{- 1}$

Step 5.1.2.2

Divide $\frac{x}{3}$ by $1$ .

$\frac{x}{3} \leq \frac{- 1}{- 1}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $- 1$ by $- 1$ .

$\frac{x}{3} \leq 1$

Step 5.2

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 \leq 1 \cdot 3$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $3$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 \leq 1 \cdot 3$

Step 5.3.1.1.2

Rewrite the expression.

$x \leq 1 \cdot 3$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $3$ by $1$ .

$x \leq 3$

Step 6

Set the argument in $\arcsin (- \frac{x}{3})$ less than or equal to $1$ to find where the expression is defined.

$- \frac{x}{3} \leq 1$

Step 7

Solve for $x$ .

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Step 7.1

Divide each term in $- \frac{x}{3} \leq 1$ by $- 1$ and simplify.

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Step 7.1.1

Divide each term in $- \frac{x}{3} \leq 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{3}}{- 1} \geq \frac{1}{- 1}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{3}}{1} \geq \frac{1}{- 1}$

Step 7.1.2.2

Divide $\frac{x}{3}$ by $1$ .

$\frac{x}{3} \geq \frac{1}{- 1}$

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $1$ by $- 1$ .

$\frac{x}{3} \geq - 1$

Step 7.2

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 \geq - 1 \cdot 3$

Step 7.3

Simplify.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Cancel the common factor of $3$ .

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Step 7.3.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 \geq - 1 \cdot 3$

Step 7.3.1.1.2

Rewrite the expression.

$x \geq - 1 \cdot 3$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Multiply $- 1$ by $3$ .

$x \geq - 3$

Step 8

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 3, 3]$

Set-Builder Notation:

${x | - 3 \leq x \leq 3}$

Step 9