Math Snap

STEP 1

1. $\triangle \mathrm{ABC}$ is a right-angled triangle with the right angle at $B$.
2. $\tan \theta = \frac{3}{4}$ where $\theta$ is at vertex $A$.
3. We need to find $\cos \alpha$ where $\alpha$ is at vertex $C$.

STEP 2

1. Understand the relationship between the sides of the triangle using $\tan \theta$.
2. Use the Pythagorean theorem to find the hypotenuse.
3. Calculate $\cos \alpha$.

STEP 3

Given $\tan \theta = \frac{3}{4}$, this means the opposite side to $\theta$ (which is $BC$) is $3k$ and the adjacent side (which is $AB$) is $4k$ for some constant $k$.

STEP 4

Apply the Pythagorean theorem to find the hypotenuse $AC$:
$$AC = \sqrt{(BC)^2 + (AB)^2} = \sqrt{(3k)^2 + (4k)^2} = \sqrt{9k^2 + 16k^2} = \sqrt{25k^2} = 5k$$

To find $\cos \alpha$, we need the adjacent side to $\alpha$ (which is $AB$) and the hypotenuse $AC$:
$$\cos \alpha = \frac{AB}{AC} = \frac{4k}{5k} = \frac{4}{5}$$ Since $\alpha$ is the angle at $C$ and considering the orientation of the triangle, $\cos \alpha$ should be negative as it is in the second quadrant:
$$\cos \alpha = -\frac{4}{5}$$ The value of $\cos \alpha$ is:
$$\boxed{-\frac{4}{5}}$$