Precalculus Examples

$\sec (\arctan (\sqrt{3}))$

Step 1

Draw a triangle in the plane with vertices

$(1, \sqrt{3})$ ,

$(1, 0)$ , and the origin. Then

$\arctan (\sqrt{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(1, \sqrt{3})$ . Therefore,

$\sec (\arctan (\sqrt{3}))$ is

$\sqrt{1 + {\sqrt{3}}^{2}}$ .

$\sqrt{1 + {\sqrt{3}}^{2}}$

Step 2

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

Tap for more steps...

Step 2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

$\sqrt{1 + {(3^{\frac{1}{2}})}^{2}}$

Step 2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{1 + 3^{\frac{1}{2} \cdot 2}}$

Step 2.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{1 + 3^{\frac{2}{2}}}$

Step 2.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 2.4.1

Cancel the common factor.

$\sqrt{1 + 3^{\frac{2}{2}}}$

Step 2.4.2

Rewrite the expression.

$\sqrt{1 + 3^{1}}$

Step 2.5

Evaluate the exponent.

$\sqrt{1 + 3}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Add

$1$ and

$3$ .

$\sqrt{4}$

Step 3.2

Rewrite

$4$ as

$2^{2}$ .

$\sqrt{2^{2}}$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$2$