Find the Other Trig Values in Quadrant I sin(0)=5/13

$\sin (0) = \frac{5}{13}$

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.

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

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.

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

Replace the known values in the equation.

$Adjacent = \sqrt{{(13)}^{2} - {(5)}^{2}}$

Simplify $\sqrt{{(13)}^{2} - {(5)}^{2}}$ .

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

Adjacent $= \sqrt{169 - {(5)}^{2}}$

Raise $5$ to the power of $2$ .

Adjacent $= \sqrt{169 - 1 \cdot 25}$

Multiply $- 1$ by $25$ .

Adjacent $= \sqrt{169 - 25}$

Subtract $25$ from $169$ .

Adjacent $= \sqrt{144}$

Rewrite $144$ as $12^{2}$ .

Adjacent $= \sqrt{12^{2}}$

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

Adjacent $= 12$

Find the value of cosine.

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

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