Trigonometry Examples

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

Step 1

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

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

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

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.

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

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

$\sqrt{{(- 1)}^{2} {(\frac{15}{17})}^{2} + {(\frac{8}{17})}^{2}}$

Step 2.2.2

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

$\sqrt{{(- 1)}^{2} \frac{15^{2}}{17^{2}} + {(\frac{8}{17})}^{2}}$

Step 2.3

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

$\sqrt{1 \frac{15^{2}}{17^{2}} + {(\frac{8}{17})}^{2}}$

Step 2.4

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

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

Step 2.5

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

$\sqrt{\frac{225}{17^{2}} + {(\frac{8}{17})}^{2}}$

Step 2.6

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

$\sqrt{\frac{225}{289} + {(\frac{8}{17})}^{2}}$

Step 2.7

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

$\sqrt{\frac{225}{289} + \frac{8^{2}}{17^{2}}}$

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Combine the numerators over the common denominator.

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

Step 2.11

Add $225$ and $64$ .

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

Step 2.12

Divide $289$ by $289$ .

$\sqrt{1}$

Step 2.13

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{8}{17}}{1}$ .

$\frac{\frac{8}{17}}{1}$

Step 4

Divide $\frac{8}{17}$ by $1$ .

$\sin (θ) = \frac{8}{17}$

Step 5

Approximate the result.

$\sin (θ) = \frac{8}{17} \approx 0.47058823$