Trigonometry Examples

$(\frac{2 \sqrt{30}}{11}, \frac{1}{11})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{2 \sqrt{30}}{11}, \frac{1}{11})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{2 \sqrt{30}}{11}, 0)$ , and $(\frac{2 \sqrt{30}}{11}, \frac{1}{11})$ .

Opposite : $\frac{1}{11}$

Adjacent : $\frac{2 \sqrt{30}}{11}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Apply the product rule to $\frac{2 \sqrt{30}}{11}$ .

$\sqrt{\frac{{(2 \sqrt{30})}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.1.2

Apply the product rule to $2 \sqrt{30}$ .

$\sqrt{\frac{2^{2} {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$\sqrt{\frac{4 {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

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Step 2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$\sqrt{\frac{4 {(30^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.2

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

$\sqrt{\frac{4 \cdot 30^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.3

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

$\sqrt{\frac{4 \cdot 30^{\frac{2}{2}}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{4 \cdot 30^{\frac{2}{2}}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{4 \cdot 30^{1}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.5

Evaluate the exponent.

$\sqrt{\frac{4 \cdot 30}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise $11$ to the power of $2$ .

$\sqrt{\frac{4 \cdot 30}{121} + {(\frac{1}{11})}^{2}}$

Step 2.3.2

Multiply $4$ by $30$ .

$\sqrt{\frac{120}{121} + {(\frac{1}{11})}^{2}}$

Step 2.3.3

Apply the product rule to $\frac{1}{11}$ .

$\sqrt{\frac{120}{121} + \frac{1^{2}}{11^{2}}}$

Step 2.3.4

One to any power is one.

$\sqrt{\frac{120}{121} + \frac{1}{11^{2}}}$

Step 2.3.5

Raise $11$ to the power of $2$ .

$\sqrt{\frac{120}{121} + \frac{1}{121}}$

Step 2.3.6

Combine the numerators over the common denominator.

$\sqrt{\frac{120 + 1}{121}}$

Step 2.3.7

Add $120$ and $1$ .

$\sqrt{\frac{121}{121}}$

Step 2.3.8

Divide $121$ by $121$ .

$\sqrt{1}$

Step 2.3.9

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{1}{11}}{1}$ .

$\frac{\frac{1}{11}}{1}$

Step 4

Divide $\frac{1}{11}$ by $1$ .

$\sin (θ) = \frac{1}{11}$

Step 5

Approximate the result.

$\sin (θ) = \frac{1}{11} \approx 0.\overline{09}$