1. **State the problem:** We have a right triangle with sides 9, 40, and 41. We need to find $\tan X$, $\cos X$, and $\sin X$ where $X$ is the angle at vertex $X$. 2. **Identify sides relative to angle $X$:** - Opposite side to $X$ is $ZY = 9$. - Adjacent side to $X$ is $XZ = 40$. - Hypotenuse is $XY = 41$. 3. **Recall trigonometric definitions:** $$\tan X = \frac{\text{opposite}}{\text{adjacent}}, \quad \cos X = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \sin X = \frac{\text{opposite}}{\text{hypotenuse}}$$ 4. **Calculate $\tan X$:** $$\tan X = \frac{9}{40}$$ 5. **Calculate $\cos X$:** $$\cos X = \frac{40}{41}$$ 6. **Calculate $\sin X$:** $$\sin X = \frac{9}{41}$$ 7. **Summary:** - $\tan X = \frac{9}{40}$ - $\cos X = \frac{40}{41}$ - $\sin X = \frac{9}{41}$