Find the Tangent Given the Point (( square root of 19)/10,9/10)

$(\frac{\sqrt{19}}{10}, \frac{9}{10})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{19}}{10}, \frac{9}{10})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{19}}{10}, 0)$ , and $(\frac{\sqrt{19}}{10}, \frac{9}{10})$ .

Opposite : $\frac{9}{10}$

Adjacent : $\frac{\sqrt{19}}{10}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{9}{10}}{\frac{\sqrt{19}}{10}}$ .

$\frac{\frac{9}{10}}{\frac{\sqrt{19}}{10}}$

Step 3

Simplify $\tan (θ)$ .

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{9}{10} \cdot \frac{10}{\sqrt{19}}$

Cancel the common factor of $10$ .

Cancel the common factor.

$\tan (θ) = \frac{9}{10} \cdot \frac{10}{\sqrt{19}}$

Rewrite the expression.

$\tan (θ) = 9 (\frac{1}{\sqrt{19}})$

Combine $9$ and $\frac{1}{\sqrt{19}}$ .

$\tan (θ) = \frac{9}{\sqrt{19}}$

Multiply $\frac{9}{\sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\tan (θ) = \frac{9}{\sqrt{19}} \cdot \frac{\sqrt{19}}{\sqrt{19}}$

Combine and simplify the denominator.

Multiply $\frac{9}{\sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\tan (θ) = \frac{9 \sqrt{19}}{\sqrt{19} \sqrt{19}}$

Raise $\sqrt{19}$ to the power of $1$ .

$\tan (θ) = \frac{9 \sqrt{19}}{\sqrt{19} \sqrt{19}}$

Raise $\sqrt{19}$ to the power of $1$ .

$\tan (θ) = \frac{9 \sqrt{19}}{\sqrt{19} \sqrt{19}}$

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

$\tan (θ) = \frac{9 \sqrt{19}}{{\sqrt{19}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (θ) = \frac{9 \sqrt{19}}{{\sqrt{19}}^{2}}$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\tan (θ) = \frac{9 \sqrt{19}}{{(19^{\frac{1}{2}})}^{2}}$

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

$\tan (θ) = \frac{9 \sqrt{19}}{19^{\frac{1}{2} \cdot 2}}$

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

$\tan (θ) = \frac{9 \sqrt{19}}{19^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (θ) = \frac{9 \sqrt{19}}{19^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (θ) = \frac{9 \sqrt{19}}{19}$

Evaluate the exponent.

$\tan (θ) = \frac{9 \sqrt{19}}{19}$

Step 4

Approximate the result.

$\tan (θ) = \frac{9 \sqrt{19}}{19} \approx 2.0647416$

Find the Tangent Given the Point (( square root of 19)/10,9/10)

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