Trigonometry Examples

$(3, - 4)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(3, - 4)$ , draw the triangle between the three points $(0, 0)$ , $(3, 0)$ , and $(3, - 4)$ .

Opposite : $- 4$

Adjacent : $3$

Step 2

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

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Step 2.1

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(- 4)}^{2}}$

Step 2.2

Raise $- 4$ to the power of $2$ .

$\sqrt{9 + 16}$

Step 2.3

Add $9$ and $16$ .

$\sqrt{25}$

Step 2.4

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}}$

Step 2.5

Pull terms out from under the radical, assuming positive real numbers.

$5$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 4}{5}$ .

$\frac{- 4}{5}$

Step 4

Move the negative in front of the fraction.

$\sin (θ) = - \frac{4}{5}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{4}{5} \approx - 0.8$