Trigonometry Examples

$(- \sqrt{11}, - 5)$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \sqrt{11}, - 5)$ , draw the triangle between the three points $(0, 0)$ , $(- \sqrt{11}, 0)$ , and $(- \sqrt{11}, - 5)$ .

Opposite : $- 5$

Adjacent : $- \sqrt{11}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{- 5}{- \sqrt{11}}$ .

$\frac{- 5}{- \sqrt{11}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Dividing two negative values results in a positive value.

$\tan (θ) = \frac{5}{\sqrt{11}}$

Step 3.2

Multiply $\frac{5}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\tan (θ) = \frac{5}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{5}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\tan (θ) = \frac{5 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 3.3.2

Raise $\sqrt{11}$ to the power of $1$ .

$\tan (θ) = \frac{5 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 3.3.3

Raise $\sqrt{11}$ to the power of $1$ .

$\tan (θ) = \frac{5 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 3.3.4

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

$\tan (θ) = \frac{5 \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 3.3.5

Add $1$ and $1$ .

$\tan (θ) = \frac{5 \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 3.3.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\tan (θ) = \frac{5 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 3.3.6.2

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

$\tan (θ) = \frac{5 \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 3.3.6.3

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

$\tan (θ) = \frac{5 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

$\tan (θ) = \frac{5 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 3.3.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{5 \sqrt{11}}{11}$

Step 3.3.6.5

Evaluate the exponent.

$\tan (θ) = \frac{5 \sqrt{11}}{11}$

Step 4

Approximate the result.

$\tan (θ) = \frac{5 \sqrt{11}}{11} \approx 1.50755672$