Find the Other Trig Values in Quadrant I sin(A)=4/5

$\sin (A) = \frac{4}{5}$

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.

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

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.

$Adjacent = \sqrt{{(5)}^{2} - {(4)}^{2}}$

Simplify inside the radical.

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

Adjacent $= \sqrt{25 - {(4)}^{2}}$

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

Adjacent $= \sqrt{25 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

Adjacent $= \sqrt{25 - 16}$

Subtract $16$ from $25$ .

Adjacent $= \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

Adjacent $= \sqrt{3^{2}}$

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

Adjacent $= 3$

Find the value of cosine.

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

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