Trigonometry Examples

$(- \frac{3}{7}, \frac{2 \sqrt{10}}{7})$

Step 1

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{3}{7}, \frac{2 \sqrt{10}}{7})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{3}{7}, 0)$ , and $(- \frac{3}{7}, \frac{2 \sqrt{10}}{7})$ .

Opposite : $\frac{2 \sqrt{10}}{7}$

Adjacent : $- \frac{3}{7}$

Step 2

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{- \frac{3}{7}}{\frac{2 \sqrt{10}}{7}}$ .

$\frac{- \frac{3}{7}}{\frac{2 \sqrt{10}}{7}}$

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = - \frac{3}{7} \cdot \frac{7}{2 \sqrt{10}}$

Step 3.2

Cancel the common factor of $7$ .

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Step 3.2.1

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$\cot (θ) = \frac{- 3}{7} \cdot \frac{7}{2 \sqrt{10}}$

Step 3.2.2

Cancel the common factor.

$\cot (θ) = \frac{- 3}{7} \cdot \frac{7}{2 \sqrt{10}}$

Step 3.2.3

Rewrite the expression.

$\cot (θ) = - 3 \frac{1}{2 \sqrt{10}}$

Step 3.3

Combine $- 3$ and $\frac{1}{2 \sqrt{10}}$ .

$\cot (θ) = \frac{- 3}{2 \sqrt{10}}$

Step 3.4

Move the negative in front of the fraction.

$\cot (θ) = - \frac{3}{2 \sqrt{10}}$

Step 3.5

Multiply $\frac{3}{2 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cot (θ) = - (\frac{3}{2 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

Multiply $\frac{3}{2 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{2 \sqrt{10} \sqrt{10}}$

Step 3.6.2

Move $\sqrt{10}$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{2 (\sqrt{10} \sqrt{10})}$

Step 3.6.3

Raise $\sqrt{10}$ to the power of $1$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{2 (\sqrt{10} \sqrt{10})}$

Step 3.6.4

Raise $\sqrt{10}$ to the power of $1$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{2 (\sqrt{10} \sqrt{10})}$

Step 3.6.5

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

$\cot (θ) = - \frac{3 \sqrt{10}}{2 {\sqrt{10}}^{1 + 1}}$

Step 3.6.6

Add $1$ and $1$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{2 {\sqrt{10}}^{2}}$

Step 3.6.7

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 3.6.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{2 {(10^{\frac{1}{2}})}^{2}}$

Step 3.6.7.2

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

$\cot (θ) = - \frac{3 \sqrt{10}}{2 \cdot 10^{\frac{1}{2} \cdot 2}}$

Step 3.6.7.3

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

$\cot (θ) = - \frac{3 \sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}$

Step 3.6.7.4

Cancel the common factor of $2$ .

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Step 3.6.7.4.1

Cancel the common factor.

$\cot (θ) = - \frac{3 \sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}$

Step 3.6.7.4.2

Rewrite the expression.

$\cot (θ) = - \frac{3 \sqrt{10}}{2 \cdot 10}$

Step 3.6.7.5

Evaluate the exponent.

$\cot (θ) = - \frac{3 \sqrt{10}}{2 \cdot 10}$

Step 3.7

Multiply $2$ by $10$ .

$\cot (θ) = - \frac{3 \sqrt{10}}{20}$

Step 4

Approximate the result.

$\cot (θ) = - \frac{3 \sqrt{10}}{20} \approx - 0.47434164$