Trigonometry Examples

$(- \frac{11}{61}, - \frac{60}{61})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{11}{61}, - \frac{60}{61})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{11}{61}, 0)$ , and $(- \frac{11}{61}, - \frac{60}{61})$ .

Opposite : $- \frac{60}{61}$

Adjacent : $- \frac{11}{61}$

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 $- \frac{11}{61}$ .

$\sqrt{{(- 1)}^{2} {(\frac{11}{61})}^{2} + {(- \frac{60}{61})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{11}{61}$ .

$\sqrt{{(- 1)}^{2} \frac{11^{2}}{61^{2}} + {(- \frac{60}{61})}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{11^{2}}{61^{2}} + {(- \frac{60}{61})}^{2}}$

Step 2.3

Multiply $\frac{11^{2}}{61^{2}}$ by $1$ .

$\sqrt{\frac{11^{2}}{61^{2}} + {(- \frac{60}{61})}^{2}}$

Step 2.4

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

$\sqrt{\frac{121}{61^{2}} + {(- \frac{60}{61})}^{2}}$

Step 2.5

Raise $61$ to the power of $2$ .

$\sqrt{\frac{121}{3721} + {(- \frac{60}{61})}^{2}}$

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 $- \frac{60}{61}$ .

$\sqrt{\frac{121}{3721} + {(- 1)}^{2} {(\frac{60}{61})}^{2}}$

Step 2.6.2

Apply the product rule to $\frac{60}{61}$ .

$\sqrt{\frac{121}{3721} + {(- 1)}^{2} \frac{60^{2}}{61^{2}}}$

Step 2.7

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

$\sqrt{\frac{121}{3721} + 1 \frac{60^{2}}{61^{2}}}$

Step 2.8

Multiply $\frac{60^{2}}{61^{2}}$ by $1$ .

$\sqrt{\frac{121}{3721} + \frac{60^{2}}{61^{2}}}$

Step 2.9

Raise $60$ to the power of $2$ .

$\sqrt{\frac{121}{3721} + \frac{3600}{61^{2}}}$

Step 2.10

Raise $61$ to the power of $2$ .

$\sqrt{\frac{121}{3721} + \frac{3600}{3721}}$

Step 2.11

Combine the numerators over the common denominator.

$\sqrt{\frac{121 + 3600}{3721}}$

Step 2.12

Add $121$ and $3600$ .

$\sqrt{\frac{3721}{3721}}$

Step 2.13

Divide $3721$ by $3721$ .

$\sqrt{1}$

Step 2.14

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{60}{61}}{1}$ .

$\frac{- \frac{60}{61}}{1}$

Step 4

Divide $- \frac{60}{61}$ by $1$ .

$\sin (θ) = - \frac{60}{61}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{60}{61} \approx - 0.98360655$