Trigonometry Examples

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

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 $\tan (θ)$ .

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Step 3.1

Dividing two negative values results in a positive value.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\tan (θ) = 2 (\frac{1}{\sqrt{21}})$

Step 3.4

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

$\tan (θ) = \frac{2}{\sqrt{21}}$

Step 3.5

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

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

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

Add $1$ and $1$ .

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

Step 3.6.6

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

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Step 3.6.6.1

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

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

Step 3.6.6.2

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

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

Step 3.6.6.3

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

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

Step 3.6.6.4

Cancel the common factor of $2$ .

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Step 3.6.6.4.1

Cancel the common factor.

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

Step 3.6.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{2 \sqrt{21}}{21}$

Step 3.6.6.5

Evaluate the exponent.

$\tan (θ) = \frac{2 \sqrt{21}}{21}$

Step 4

Approximate the result.

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