Find the Other Trig Values in Quadrant I csc(0)=5/3

$\csc (0) = \frac{5}{3}$

Step 1

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

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

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.

$Adjacent = \sqrt{{(5)}^{2} - {(3)}^{2}}$

Step 4

Simplify inside the radical.

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

Adjacent $= \sqrt{25 - {(3)}^{2}}$

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

Adjacent $= \sqrt{25 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

Adjacent $= \sqrt{25 - 9}$

Subtract $9$ from $25$ .

Adjacent $= \sqrt{16}$

Rewrite $16$ as $4^{2}$ .

Adjacent $= \sqrt{4^{2}}$

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

Adjacent $= 4$

Step 5

Find the value of sine.

Use the definition of sine to find the value of $\sin (0)$ .

$\sin (0) = \frac{opp}{hyp}$