Trigonometry Examples

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

Step 1

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

$x = \sin (\arctan (\frac{1}{4}))$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Draw a triangle in the plane with vertices $(1, \frac{1}{4})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{1}{4})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{1}{4})$ . Therefore, $\sin (\arctan (\frac{1}{4}))$ is $\frac{\sqrt{17}}{17}$ .

Tap for more steps...

Step 2.1.1

Multiply $\frac{1}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$x = \frac{\sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 2.1.2

Raise $\sqrt{17}$ to the power of $1$ .

$x = \frac{\sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}}$

Step 2.1.3

Raise $\sqrt{17}$ to the power of $1$ .

$x = \frac{\sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}}$

Step 2.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{\sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Step 2.1.5

Add $1$ and $1$ .

$x = \frac{\sqrt{17}}{{\sqrt{17}}^{2}}$

Step 2.1.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

Tap for more steps...

Step 2.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$x = \frac{\sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Step 2.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{\sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Step 2.1.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{\sqrt{17}}{17^{\frac{2}{2}}}$

Step 2.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.6.4.1

Cancel the common factor.

$x = \frac{\sqrt{17}}{17^{\frac{2}{2}}}$

Step 2.1.6.4.2

Rewrite the expression.

$x = \frac{\sqrt{17}}{17^{1}}$

Step 2.1.6.5

Evaluate the exponent.

$x = \frac{\sqrt{17}}{17}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{17}}{17}$

Decimal Form:

$x = 0.24253562 \dots$