Trigonometry Examples

$(- \sqrt{5}, 1)$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \sqrt{5}, 1)$ , draw the triangle between the three points $(0, 0)$ , $(- \sqrt{5}, 0)$ , and $(- \sqrt{5}, 1)$ .

Opposite : $1$

Adjacent : $- \sqrt{5}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{1}{- \sqrt{5}}$ .

$\frac{1}{- \sqrt{5}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.1.2

Move the negative in front of the fraction.

$\tan (θ) = - \frac{1}{\sqrt{5}}$

Step 3.2

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 3.3.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 3.3.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

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.

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

Step 3.3.6.4.2

Rewrite the expression.

$\tan (θ) = - \frac{\sqrt{5}}{5}$

Step 3.3.6.5

Evaluate the exponent.

$\tan (θ) = - \frac{\sqrt{5}}{5}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{\sqrt{5}}{5} \approx - 0.44721359$