Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

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 1.2

Apply the product rule to $- \sqrt{3}$ .

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

Step 1.3

Raise $- 1$ to the power of $2$ .

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

Step 1.4

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

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

Step 2

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

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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 $2$ .

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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.

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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.

$2$