Trigonometry Examples

$(\frac{\sqrt{77}}{9}, \frac{2}{9})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{77}}{9}, \frac{2}{9})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{77}}{9}, 0)$ , and $(\frac{\sqrt{77}}{9}, \frac{2}{9})$ .

Opposite : $\frac{2}{9}$

Adjacent : $\frac{\sqrt{77}}{9}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{2}{9}}{\frac{\sqrt{77}}{9}}$ .

$\frac{\frac{2}{9}}{\frac{\sqrt{77}}{9}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{2}{9} \cdot \frac{9}{\sqrt{77}}$

Step 3.2

Cancel the common factor of $9$ .

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Step 3.2.1

Cancel the common factor.

$\tan (θ) = \frac{2}{9} \cdot \frac{9}{\sqrt{77}}$

Step 3.2.2

Rewrite the expression.

$\tan (θ) = 2 (\frac{1}{\sqrt{77}})$

Step 3.3

Combine $2$ and $\frac{1}{\sqrt{77}}$ .

$\tan (θ) = \frac{2}{\sqrt{77}}$

Step 3.4

Multiply $\frac{2}{\sqrt{77}}$ by $\frac{\sqrt{77}}{\sqrt{77}}$ .

$\tan (θ) = \frac{2}{\sqrt{77}} \cdot \frac{\sqrt{77}}{\sqrt{77}}$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{2}{\sqrt{77}}$ by $\frac{\sqrt{77}}{\sqrt{77}}$ .

$\tan (θ) = \frac{2 \sqrt{77}}{\sqrt{77} \sqrt{77}}$

Step 3.5.2

Raise $\sqrt{77}$ to the power of $1$ .

$\tan (θ) = \frac{2 \sqrt{77}}{\sqrt{77} \sqrt{77}}$

Step 3.5.3

Raise $\sqrt{77}$ to the power of $1$ .

$\tan (θ) = \frac{2 \sqrt{77}}{\sqrt{77} \sqrt{77}}$

Step 3.5.4

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

$\tan (θ) = \frac{2 \sqrt{77}}{{\sqrt{77}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$\tan (θ) = \frac{2 \sqrt{77}}{{\sqrt{77}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{77}}^{2}$ as $77$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{77}$ as $77^{\frac{1}{2}}$ .

$\tan (θ) = \frac{2 \sqrt{77}}{{(77^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$\tan (θ) = \frac{2 \sqrt{77}}{77^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$\tan (θ) = \frac{2 \sqrt{77}}{77^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$\tan (θ) = \frac{2 \sqrt{77}}{77^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{2 \sqrt{77}}{77}$

Step 3.5.6.5

Evaluate the exponent.

$\tan (θ) = \frac{2 \sqrt{77}}{77}$

Step 4

Approximate the result.

$\tan (θ) = \frac{2 \sqrt{77}}{77} \approx 0.22792115$