Trigonometry Examples

$(- \frac{\sqrt{7}}{4}, \frac{3}{4})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{7}}{4}, \frac{3}{4})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{7}}{4}, 0)$ , and $(- \frac{\sqrt{7}}{4}, \frac{3}{4})$ .

Opposite : $\frac{3}{4}$

Adjacent : $- \frac{\sqrt{7}}{4}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{3}{4}}{- \frac{\sqrt{7}}{4}}$ .

$\frac{\frac{3}{4}}{- \frac{\sqrt{7}}{4}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{3}{4} \cdot (- \frac{4}{\sqrt{7}})$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Move the leading negative in $- \frac{4}{\sqrt{7}}$ into the numerator.

$\tan (θ) = \frac{3}{4} \cdot \frac{- 4}{\sqrt{7}}$

Step 3.2.2

Factor $4$ out of $- 4$ .

$\tan (θ) = \frac{3}{4} \cdot \frac{4 (- 1)}{\sqrt{7}}$

Step 3.2.3

Cancel the common factor.

$\tan (θ) = \frac{3}{4} \cdot \frac{4 \cdot - 1}{\sqrt{7}}$

Step 3.2.4

Rewrite the expression.

$\tan (θ) = 3 (\frac{- 1}{\sqrt{7}})$

Step 3.3

Combine $3$ and $\frac{- 1}{\sqrt{7}}$ .

$\tan (θ) = \frac{3 \cdot - 1}{\sqrt{7}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Multiply $3$ by $- 1$ .

$\tan (θ) = \frac{- 3}{\sqrt{7}}$

Step 3.4.2

Move the negative in front of the fraction.

$\tan (θ) = - \frac{3}{\sqrt{7}}$

Step 3.5

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\tan (θ) = - (\frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\tan (θ) = - \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.6.2

Raise $\sqrt{7}$ to the power of $1$ .

$\tan (θ) = - \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.6.3

Raise $\sqrt{7}$ to the power of $1$ .

$\tan (θ) = - \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.6.4

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

$\tan (θ) = - \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.6.5

Add $1$ and $1$ .

$\tan (θ) = - \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.6.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\tan (θ) = - \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.6.6.2

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

$\tan (θ) = - \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.6.6.3

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

$\tan (θ) = - \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

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.

$\tan (θ) = - \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.6.6.4.2

Rewrite the expression.

$\tan (θ) = - \frac{3 \sqrt{7}}{7}$

Step 3.6.6.5

Evaluate the exponent.

$\tan (θ) = - \frac{3 \sqrt{7}}{7}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{3 \sqrt{7}}{7} \approx - 1.13389341$