Trigonometry Examples

$(\frac{8}{17}, - \frac{15}{17})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{8}{17}, - \frac{15}{17})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{8}{17}, 0)$ , and $(\frac{8}{17}, - \frac{15}{17})$ .

Opposite : $- \frac{15}{17}$

Adjacent : $\frac{8}{17}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{8}{17}$ .

$\sqrt{\frac{8^{2}}{17^{2}} + {(- \frac{15}{17})}^{2}}$

Step 2.2

Raise $8$ to the power of $2$ .

$\sqrt{\frac{64}{17^{2}} + {(- \frac{15}{17})}^{2}}$

Step 2.3

Raise $17$ to the power of $2$ .

$\sqrt{\frac{64}{289} + {(- \frac{15}{17})}^{2}}$

Step 2.4

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

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Step 2.4.1

Apply the product rule to $- \frac{15}{17}$ .

$\sqrt{\frac{64}{289} + {(- 1)}^{2} {(\frac{15}{17})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{15}{17}$ .

$\sqrt{\frac{64}{289} + {(- 1)}^{2} \frac{15^{2}}{17^{2}}}$

Step 2.5

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

$\sqrt{\frac{64}{289} + 1 \frac{15^{2}}{17^{2}}}$

Step 2.6

Multiply $\frac{15^{2}}{17^{2}}$ by $1$ .

$\sqrt{\frac{64}{289} + \frac{15^{2}}{17^{2}}}$

Step 2.7

Raise $15$ to the power of $2$ .

$\sqrt{\frac{64}{289} + \frac{225}{17^{2}}}$

Step 2.8

Raise $17$ to the power of $2$ .

$\sqrt{\frac{64}{289} + \frac{225}{289}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{64 + 225}{289}}$

Step 2.10

Add $64$ and $225$ .

$\sqrt{\frac{289}{289}}$

Step 2.11

Divide $289$ by $289$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{15}{17}}{1}$ .

$\frac{- \frac{15}{17}}{1}$

Step 4

Divide $- \frac{15}{17}$ by $1$ .

$\sin (θ) = - \frac{15}{17}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{15}{17} \approx - 0.88235294$