Trigonometry Examples

$(\frac{\sqrt{2}}{3}, \frac{\sqrt{7}}{3})$

Step 1

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{2}}{3}, \frac{\sqrt{7}}{3})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{2}}{3}, 0)$ , and $(\frac{\sqrt{2}}{3}, \frac{\sqrt{7}}{3})$ .

Opposite : $\frac{\sqrt{7}}{3}$

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

Step 2

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{\frac{\sqrt{2}}{3}}{\frac{\sqrt{7}}{3}}$ .

$\frac{\frac{\sqrt{2}}{3}}{\frac{\sqrt{7}}{3}}$

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = \frac{\sqrt{2}}{3} \cdot \frac{3}{\sqrt{7}}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Cancel the common factor.

$\cot (θ) = \frac{\sqrt{2}}{3} \cdot \frac{3}{\sqrt{7}}$

Step 3.2.2

Rewrite the expression.

$\cot (θ) = \sqrt{2} (\frac{1}{\sqrt{7}})$

Step 3.3

Combine $\sqrt{2}$ and $\frac{1}{\sqrt{7}}$ .

$\cot (θ) = \frac{\sqrt{2}}{\sqrt{7}}$

Step 3.4

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (θ) = \frac{\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.5.4

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

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{7}$

Step 3.5.6.5

Evaluate the exponent.

$\cot (θ) = \frac{\sqrt{2} \sqrt{7}}{7}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Combine using the product rule for radicals.

$\cot (θ) = \frac{\sqrt{2 \cdot 7}}{7}$

Step 3.6.2

Multiply $2$ by $7$ .

$\cot (θ) = \frac{\sqrt{14}}{7}$

Step 4

Approximate the result.

$\cot (θ) = \frac{\sqrt{14}}{7} \approx 0.53452248$