Trigonometry Examples

$- 6 \arcsin (2 x) = 2 π$

Step 1

Divide each term in $- 6 \arcsin (2 x) = 2 π$ by $- 6$ and simplify.

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Step 1.1

Divide each term in $- 6 \arcsin (2 x) = 2 π$ by $- 6$ .

$\frac{- 6 \arcsin (2 x)}{- 6} = \frac{2 π}{- 6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 6 \arcsin (2 x)}{- 6} = \frac{2 π}{- 6}$

Step 1.2.1.2

Divide $\arcsin (2 x)$ by $1$ .

$\arcsin (2 x) = \frac{2 π}{- 6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2$ and $- 6$ .

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Step 1.3.1.1

Factor $2$ out of $2 π$ .

$\arcsin (2 x) = \frac{2 (π)}{- 6}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $- 6$ .

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

Step 1.3.1.2.2

Cancel the common factor.

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

Step 1.3.1.2.3

Rewrite the expression.

$\arcsin (2 x) = \frac{π}{- 3}$

Step 1.3.2

Move the negative in front of the fraction.

$\arcsin (2 x) = - \frac{π}{3}$

Step 2

Take the inverse arcsine of both sides of the equation to extract $x$ from inside the arcsine.

$2 x = \sin (- \frac{π}{3})$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\sin (- \frac{π}{3})$ .

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Step 3.1.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 x = \sin (\frac{5 π}{3})$

Step 3.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$2 x = - \sin (\frac{π}{3})$

Step 3.1.3

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$2 x = - \frac{\sqrt{3}}{2}$

Step 4

Divide each term in $2 x = - \frac{\sqrt{3}}{2}$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = - \frac{\sqrt{3}}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{\sqrt{3}}{2}}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{\sqrt{3}}{2}}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{\sqrt{3}}{2}}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{\sqrt{3}}{2} \cdot \frac{1}{2}$

Step 4.3.2

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

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Step 4.3.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3}}{2}$ .

$x = - \frac{\sqrt{3}}{2 \cdot 2}$

Step 4.3.2.2

Multiply $2$ by $2$ .

$x = - \frac{\sqrt{3}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\sqrt{3}}{4}$

Decimal Form:

$x = - 0.43301270 \dots$