Find the Other Trig Values in Quadrant I cos(b)=4/5

$\cos (b) = \frac{4}{5}$

Step 1

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

$\cos (b) = \frac{adjacent}{hypotenuse}$

Step 2

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

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

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

Step 4

Simplify inside the radical.

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

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

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

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

Multiply $- 1$ by $16$ .

Opposite $= \sqrt{25 - 16}$

Subtract $16$ from $25$ .

Opposite $= \sqrt{9}$

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

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

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

Opposite $= 3$

Step 5

Find the value of sine.

Use the definition of sine to find the value of $\sin (b)$ .

$\sin (b) = \frac{opp}{hyp}$