Trigonometry Examples

$(- 2 \sqrt{3}, 4)$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2 \sqrt{3}, 4)$ , draw the triangle between the three points $(0, 0)$ , $(- 2 \sqrt{3}, 0)$ , and $(- 2 \sqrt{3}, 4)$ .

Opposite : $4$

Adjacent : $- 2 \sqrt{3}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{4}{- 2 \sqrt{3}}$ .

$\frac{4}{- 2 \sqrt{3}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Cancel the common factor of $4$ and $- 2$ .

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Step 3.1.1

Factor $2$ out of $4$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

$\tan (θ) = \frac{2}{- \sqrt{3}}$

Step 3.2

Move the negative in front of the fraction.

$\tan (θ) = - \frac{2}{\sqrt{3}}$

Step 3.3

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

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.4.6

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

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Step 3.4.6.1

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

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

Step 3.4.6.2

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

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

Step 3.4.6.3

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

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

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

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

Step 3.4.6.4.2

Rewrite the expression.

$\tan (θ) = - \frac{2 \sqrt{3}}{3}$

Step 3.4.6.5

Evaluate the exponent.

$\tan (θ) = - \frac{2 \sqrt{3}}{3}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{2 \sqrt{3}}{3} \approx - 1.15470053$