Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

Draw a triangle in the plane with vertices $(1, \frac{\sqrt{3}}{3})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{\sqrt{3}}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{\sqrt{3}}{3})$ . Therefore, $\sec (\arctan (\frac{\sqrt{3}}{3}))$ is $\sqrt{1 + {(\frac{\sqrt{3}}{3})}^{2}}$ .

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

Step 1.2

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

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Evaluate the exponent.

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

Step 3

Raise $3$ to the power of $2$ .

$\sqrt{1 + \frac{3}{9}}$

Step 4

Cancel the common factor of $3$ and $9$ .

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Step 4.1

Factor $3$ out of $3$ .

$\sqrt{1 + \frac{3 (1)}{9}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $9$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Simplify the expression.

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Step 5.1

Write $1$ as a fraction with a common denominator.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Add $3$ and $1$ .

$\sqrt{\frac{4}{3}}$

Step 6

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$\frac{\sqrt{4}}{\sqrt{3}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.2

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

$\frac{2}{\sqrt{3}}$

Step 8

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

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

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

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 9.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

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

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

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

Step 9.6.5

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3}$

Decimal Form:

$1.15470053 \dots$