Trigonometry Examples

$- \arcsin (4 x) = \frac{π}{4}$

Step 1

Divide each term in $- \arcsin (4 x) = \frac{π}{4}$ by $- 1$ and simplify.

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Step 1.1

Divide each term in $- \arcsin (4 x) = \frac{π}{4}$ by $- 1$ .

$\frac{- \arcsin (4 x)}{- 1} = \frac{\frac{π}{4}}{- 1}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Dividing two negative values results in a positive value.

$\frac{\arcsin (4 x)}{1} = \frac{\frac{π}{4}}{- 1}$

Step 1.2.2

Divide $\arcsin (4 x)$ by $1$ .

$\arcsin (4 x) = \frac{\frac{π}{4}}{- 1}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative one from the denominator of $\frac{\frac{π}{4}}{- 1}$ .

$\arcsin (4 x) = - 1 \cdot \frac{π}{4}$

Step 1.3.2

Rewrite $- 1 \cdot \frac{π}{4}$ as $- \frac{π}{4}$ .

$\arcsin (4 x) = - \frac{π}{4}$

Step 2

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

$4 x = \sin (- \frac{π}{4})$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\sin (- \frac{π}{4})$ .

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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 $2 π$ .

$4 x = \sin (\frac{7 π}{4})$

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.

$4 x = - \sin (\frac{π}{4})$

Step 3.1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$4 x = - \frac{\sqrt{2}}{2}$

Step 4

Divide each term in $4 x = - \frac{\sqrt{2}}{2}$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 x = - \frac{\sqrt{2}}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{- \frac{\sqrt{2}}{2}}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \frac{\sqrt{2}}{2}}{4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{\sqrt{2}}{2}}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{\sqrt{2}}{2} \cdot \frac{1}{4}$

Step 4.3.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{4}$ .

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Step 4.3.2.1

Multiply $\frac{1}{4}$ by $\frac{\sqrt{2}}{2}$ .

$x = - \frac{\sqrt{2}}{4 \cdot 2}$

Step 4.3.2.2

Multiply $4$ by $2$ .

$x = - \frac{\sqrt{2}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\sqrt{2}}{8}$

Decimal Form:

$x = - 0.17677669 \dots$