Question

find the value of \\(\cos t\\) rounded to the nearest hundredth, if necessary.
(image of triangle rst with right angle at s, rs=48, rt=50, st=14)
answer
\\(\cos t = \\) input box submit answer

Explanation:

Step1: Recall cosine definition

In a right triangle, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ for angle $\theta$.

Step2: Identify sides for $\angle T$

For $\angle T$, adjacent side is $14$, hypotenuse is $50$ (wait, no—wait, triangle $RST$ is right-angled at $S$. So sides: $ST = 14$, $RT = 50$? Wait, no, let's check: $RS = 48$, $ST = 14$, $RT = 50$ (since $48^2 + 14^2 = 2304 + 196 = 2500 = 50^2$, so right-angled at $S$). So for $\angle T$, the adjacent side is $ST = 14$, hypotenuse is $RT = 50$? Wait, no—wait, in angle $T$, the sides: the angle at $T$, so the sides: adjacent is $ST$ (length 14), hypotenuse is $RT$ (length 50)? Wait, no, cosine of angle $T$: in triangle $RST$, right-angled at $S$, so angle at $T$: the sides adjacent to $T$ is $ST = 14$, opposite is $RS = 48$, hypotenuse is $RT = 50$. So $\cos T = \frac{\text{adjacent to } T}{\text{hypotenuse}} = \frac{ST}{RT} = \frac{14}{50}$? Wait, no, wait: wait, in angle $T$, the adjacent side is $ST$ (length 14), and the hypotenuse is $RT$ (length 50)? Wait, no, that can't be, because $ST$ is one leg, $RT$ is the hypotenuse. Wait, let's re-express: in right triangle $RST$, right-angled at $S$, so vertices: $R$, $S$ (right angle), $T$. So sides: $RS = 48$ (leg), $ST = 14$ (leg), $RT = 50$ (hypotenuse). So angle at $T$: the sides: adjacent to $T$ is $ST$ (length 14), opposite is $RS$ (length 48), hypotenuse is $RT$ (length 50). Therefore, $\cos T = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{ST}{RT} = \frac{14}{50}$? Wait, no, that's not right. Wait, no—wait, maybe I mixed up. Wait, in angle $T$, the sides: the angle at $T$ is between $ST$ and $RT$. So the adjacent side is $ST$ (length 14), and the hypotenuse is $RT$ (length 50). Wait, but let's calculate $\frac{14}{50} = 0.28$? Wait, but that seems low. Wait, no, wait: wait, $48^2 + 14^2 = 2304 + 196 = 2500 = 50^2$, so that's correct. So $\cos T = \frac{ST}{RT} = \frac{14}{50} = 0.28$? Wait, but let's check again. Wait, maybe I got the adjacent and hypotenuse wrong. Wait, in angle $T$, the sides: the adjacent side is the one that is part of the angle and is not the hypotenuse. So angle $T$ is at vertex $T$, so the sides meeting at $T$ are $ST$ (length 14) and $RT$ (length 50). The other side is $RS$ (length 48). So in angle $T$, the adjacent side is $ST$ (length 14), hypotenuse is $RT$ (length 50), so $\cos T = \frac{ST}{RT} = \frac{14}{50} = 0.28$. Wait, but let's confirm with the cosine law. Cosine law: $RS^2 = ST^2 + RT^2 - 2 \cdot ST \cdot RT \cdot \cos T$. Wait, no, cosine law: $c^2 = a^2 + b^2 - 2ab \cos C$. So in triangle $RST$, with $RS = 48$, $ST = 14$, $RT = 50$, angle at $T$: so $RS^2 = ST^2 + RT^2 - 2 \cdot ST \cdot RT \cdot \cos T$. So $48^2 = 14^2 + 50^2 - 2 \cdot 14 \cdot 50 \cdot \cos T$. Let's compute: $2304 = 196 + 2500 - 1400 \cos T$. $2304 = 2696 - 1400 \cos T$. Then, $1400 \cos T = 2696 - 2304 = 392$. So $\cos T = \frac{392}{1400} = 0.28$. Yes, that matches. So the calculation is correct.

Answer:

$\boxed{0.28}$