Question: Find the exact value of csc $M$ in simplest form.
$M$
$\sqrt{15}$
$8$
$7$
$K$
$L$
Graph shape: a right triangle with vertices $M$ at the top, $K$ at the bottom-left, and $L$ at the right; the right angle is at $L$. The side $MK$ is labeled $8$, the side $ML$ is labeled $\sqrt{15}$, and the side $KL$ is labeled $7$. Position hint: center
1. **State the problem:** We need to find the exact value of $\csc M$ in simplest form for the right triangle $\triangle MKL$ with right angle at $L$.
2. **Identify sides relative to angle $M$:**
- Hypotenuse: side opposite the right angle $L$ is $MK = 8$.
- Opposite side to angle $M$ is $KL = 7$.
- Adjacent side to angle $M$ is $ML = \sqrt{15}$.
3. **Recall the definition of cosecant:**
$$\csc M = \frac{1}{\sin M} = \frac{\text{hypotenuse}}{\text{opposite}}$$
4. **Calculate $\csc M$ using the sides:**
$$\csc M = \frac{MK}{KL} = \frac{8}{7}$$
5. **Simplify the fraction if possible:**
$\frac{8}{7}$ is already in simplest form.
**Final answer:**
$$\boxed{\frac{8}{7}}$$