Trigonometry Examples

$(\frac{- \sqrt{11}}{6}, \frac{- 5}{6})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{- \sqrt{11}}{6}, \frac{- 5}{6})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{- \sqrt{11}}{6}, 0)$ , and $(\frac{- \sqrt{11}}{6}, \frac{- 5}{6})$ .

Opposite : $\frac{- 5}{6}$

Adjacent : $\frac{- \sqrt{11}}{6}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Move the negative in front of the fraction.

$\sqrt{{(- \frac{\sqrt{11}}{6})}^{2} + {(\frac{- 5}{6})}^{2}}$

Step 2.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1

Apply the product rule to $- \frac{\sqrt{11}}{6}$ .

$\sqrt{{(- 1)}^{2} {(\frac{\sqrt{11}}{6})}^{2} + {(\frac{- 5}{6})}^{2}}$

Step 2.2.2

Apply the product rule to $\frac{\sqrt{11}}{6}$ .

$\sqrt{{(- 1)}^{2} \frac{{\sqrt{11}}^{2}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{{\sqrt{11}}^{2}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.3.2

Multiply $\frac{{\sqrt{11}}^{2}}{6^{2}}$ by $1$ .

$\sqrt{\frac{{\sqrt{11}}^{2}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.4

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{\frac{{(11^{\frac{1}{2}})}^{2}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.4.2

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

$\sqrt{\frac{11^{\frac{1}{2} \cdot 2}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.4.3

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

$\sqrt{\frac{11^{\frac{2}{2}}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

$\sqrt{\frac{11^{\frac{2}{2}}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.4.4.2

Rewrite the expression.

$\sqrt{\frac{11^{1}}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.4.5

Evaluate the exponent.

$\sqrt{\frac{11}{6^{2}} + {(\frac{- 5}{6})}^{2}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Raise $6$ to the power of $2$ .

$\sqrt{\frac{11}{36} + {(\frac{- 5}{6})}^{2}}$

Step 2.5.2

Move the negative in front of the fraction.

$\sqrt{\frac{11}{36} + {(- \frac{5}{6})}^{2}}$

Step 2.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.6.1

Apply the product rule to $- \frac{5}{6}$ .

$\sqrt{\frac{11}{36} + {(- 1)}^{2} {(\frac{5}{6})}^{2}}$

Step 2.6.2

Apply the product rule to $\frac{5}{6}$ .

$\sqrt{\frac{11}{36} + {(- 1)}^{2} \frac{5^{2}}{6^{2}}}$

Step 2.7

Raise $- 1$ to the power of $2$ .

$\sqrt{\frac{11}{36} + 1 \frac{5^{2}}{6^{2}}}$

Step 2.8

Multiply $\frac{5^{2}}{6^{2}}$ by $1$ .

$\sqrt{\frac{11}{36} + \frac{5^{2}}{6^{2}}}$

Step 2.9

Raise $5$ to the power of $2$ .

$\sqrt{\frac{11}{36} + \frac{25}{6^{2}}}$

Step 2.10

Raise $6$ to the power of $2$ .

$\sqrt{\frac{11}{36} + \frac{25}{36}}$

Step 2.11

Combine the numerators over the common denominator.

$\sqrt{\frac{11 + 25}{36}}$

Step 2.12

Add $11$ and $25$ .

$\sqrt{\frac{36}{36}}$

Step 2.13

Divide $36$ by $36$ .

$\sqrt{1}$

Step 2.14

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{- 5}{6}}{1}$ .

$\frac{\frac{- 5}{6}}{1}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Divide $\frac{- 5}{6}$ by $1$ .

$\sin (θ) = \frac{- 5}{6}$

Step 4.2

Move the negative in front of the fraction.

$\sin (θ) = - \frac{5}{6}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{5}{6} \approx - 0.8\overline{3}$