Find the Other Trig Values in Quadrant I sin(0)=12/13

$\sin (0) = \frac{12}{13}$

Step 1

Use the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\sin (0) = \frac{opposite}{hypotenuse}$

Step 2

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(13)}^{2} - {(12)}^{2}}$

Step 4

Simplify inside the radical.

Raise $13$ to the power of $2$ .

Adjacent $= \sqrt{169 - {(12)}^{2}}$

Raise $12$ to the power of $2$ .

Adjacent $= \sqrt{169 - 1 \cdot 144}$

Multiply $- 1$ by $144$ .

Adjacent $= \sqrt{169 - 144}$

Subtract $144$ from $169$ .

Adjacent $= \sqrt{25}$

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

Adjacent $= \sqrt{5^{2}}$

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

Adjacent $= 5$

Step 5

Find the value of cosine.

Use the definition of cosine to find the value of $\cos (0)$ .

$\cos (0) = \frac{adj}{hyp}$