Find the Tangent Given the Point (-( square root of 21)/5,2/5)

$(- \frac{\sqrt{21}}{5}, \frac{2}{5})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{21}}{5}, \frac{2}{5})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{21}}{5}, 0)$ , and $(- \frac{\sqrt{21}}{5}, \frac{2}{5})$ .

Opposite : $\frac{2}{5}$

Adjacent : $- \frac{\sqrt{21}}{5}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{2}{5}}{- \frac{\sqrt{21}}{5}}$ .

$\frac{\frac{2}{5}}{- \frac{\sqrt{21}}{5}}$

Step 3

Simplify $\tan (θ)$ .

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{2}{5} \cdot (- \frac{5}{\sqrt{21}})$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{\sqrt{21}}$ into the numerator.

$\tan (θ) = \frac{2}{5} \cdot \frac{- 5}{\sqrt{21}}$

Factor $5$ out of $- 5$ .

$\tan (θ) = \frac{2}{5} \cdot \frac{5 (- 1)}{\sqrt{21}}$

Cancel the common factor.

$\tan (θ) = \frac{2}{5} \cdot \frac{5 \cdot - 1}{\sqrt{21}}$

Rewrite the expression.

$\tan (θ) = 2 (\frac{- 1}{\sqrt{21}})$

Combine $2$ and $\frac{- 1}{\sqrt{21}}$ .

$\tan (θ) = \frac{2 \cdot - 1}{\sqrt{21}}$

Simplify the expression.

Multiply $2$ by $- 1$ .

$\tan (θ) = \frac{- 2}{\sqrt{21}}$

Move the negative in front of the fraction.

$\tan (θ) = - \frac{2}{\sqrt{21}}$

Multiply $\frac{2}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\tan (θ) = - (\frac{2}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Raise $\sqrt{21}$ to the power of $1$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Raise $\sqrt{21}$ to the power of $1$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

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

$\tan (θ) = - \frac{2 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{{\sqrt{21}}^{2}}$

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

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

$\tan (θ) = - \frac{2 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

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

$\tan (θ) = - \frac{2 \sqrt{21}}{21^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (θ) = - \frac{2 \sqrt{21}}{21^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (θ) = - \frac{2 \sqrt{21}}{21}$

Evaluate the exponent.

$\tan (θ) = - \frac{2 \sqrt{21}}{21}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{2 \sqrt{21}}{21} \approx - 0.43643578$

Find the Tangent Given the Point (-( square root of 21)/5,2/5)

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