Trigonometry Examples

$(3 \frac{1}{2}, - 2 \sqrt{15})$

Step 1

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(3 \frac{1}{2}, - 2 \sqrt{15})$ , draw the triangle between the three points $(0, 0)$ , $(3 \frac{1}{2}, 0)$ , and $(3 \frac{1}{2}, - 2 \sqrt{15})$ .

Opposite : $- 2 \sqrt{15}$

Adjacent : $3 \frac{1}{2}$

Step 2

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{3 \frac{1}{2}}{- 2 \sqrt{15}}$ .

$\frac{3 \frac{1}{2}}{- 2 \sqrt{15}}$

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Convert $3 \frac{1}{2}$ to an improper fraction.

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Step 3.1.1

A mixed number is an addition of its whole and fractional parts.

$\cot (θ) = \frac{3 + \frac{1}{2}}{- 2 \sqrt{15}}$

Step 3.1.2

Add $3$ and $\frac{1}{2}$ .

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Step 3.1.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\cot (θ) = \frac{3 \cdot \frac{2}{2} + \frac{1}{2}}{- 2 \sqrt{15}}$

Step 3.1.2.2

Combine $3$ and $\frac{2}{2}$ .

$\cot (θ) = \frac{\frac{3 \cdot 2}{2} + \frac{1}{2}}{- 2 \sqrt{15}}$

Step 3.1.2.3

Combine the numerators over the common denominator.

$\cot (θ) = \frac{\frac{3 \cdot 2 + 1}{2}}{- 2 \sqrt{15}}$

Step 3.1.2.4

Simplify the numerator.

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Step 3.1.2.4.1

Multiply $3$ by $2$ .

$\cot (θ) = \frac{\frac{6 + 1}{2}}{- 2 \sqrt{15}}$

Step 3.1.2.4.2

Add $6$ and $1$ .

$\cot (θ) = \frac{\frac{7}{2}}{- 2 \sqrt{15}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = \frac{7}{2} \cdot \frac{1}{- 2 \sqrt{15}}$

Step 3.3

Move the negative in front of the fraction.

$\cot (θ) = \frac{7}{2} \cdot (- \frac{1}{2 \sqrt{15}})$

Step 3.4

Multiply $\frac{1}{2 \sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\cot (θ) = \frac{7}{2} \cdot (- (\frac{1}{2 \sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}))$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{1}{2 \sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 \sqrt{15} \sqrt{15}})$

Step 3.5.2

Move $\sqrt{15}$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 3.5.3

Raise $\sqrt{15}$ to the power of $1$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 3.5.4

Raise $\sqrt{15}$ to the power of $1$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 3.5.5

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

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 {\sqrt{15}}^{1 + 1}})$

Step 3.5.6

Add $1$ and $1$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 {\sqrt{15}}^{2}})$

Step 3.5.7

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 3.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 {(15^{\frac{1}{2}})}^{2}})$

Step 3.5.7.2

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

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15^{\frac{1}{2} \cdot 2}})$

Step 3.5.7.3

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

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15^{\frac{2}{2}}})$

Step 3.5.7.4

Cancel the common factor of $2$ .

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Step 3.5.7.4.1

Cancel the common factor.

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15^{\frac{2}{2}}})$

Step 3.5.7.4.2

Rewrite the expression.

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15})$

Step 3.5.7.5

Evaluate the exponent.

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15})$

Step 3.6

Multiply $2$ by $15$ .

$\cot (θ) = \frac{7}{2} \cdot (- \frac{\sqrt{15}}{30})$

Step 3.7

Multiply $\frac{7}{2} (- \frac{\sqrt{15}}{30})$ .

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Step 3.7.1

Multiply $\frac{7}{2}$ by $\frac{\sqrt{15}}{30}$ .

$\cot (θ) = - \frac{7 \sqrt{15}}{2 \cdot 30}$

Step 3.7.2

Multiply $2$ by $30$ .

$\cot (θ) = - \frac{7 \sqrt{15}}{60}$

Step 4

Approximate the result.

$\cot (θ) = - \frac{7 \sqrt{15}}{60} \approx - 0.45184805$