1. **Problem Statement:** We have a right triangle RST with a right angle at S. Side RT (hypotenuse) is 35, side TS is 21, and we want to determine if each expression can find the length of side RS. 2. **Recall definitions:** - Hypotenuse (opposite right angle) = RT = 35 - Opposite side to angle R = TS = 21 - Adjacent side to angle R = RS (unknown) 3. **Check each expression:** - Expression: $35\sin(R)$ - $\sin(R) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{TS}{RT} = \frac{21}{35}$ - So, $35\sin(R) = 35 \times \frac{21}{35} = 21 = TS$, not RS. - **Answer: No** - Expression: $21\tan(T)$ - $\tan(T) = \frac{\text{opposite}}{\text{adjacent}}$ - Opposite to $T$ is $RS$, adjacent to $T$ is $TS=21$ - So, $\tan(T) = \frac{RS}{21}$ - Then, $21\tan(T) = 21 \times \frac{RS}{21} = RS$ - **Answer: Yes** - Expression: $35\cos(R)$ - $\cos(R) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{RS}{35}$ - So, $35\cos(R) = 35 \times \frac{RS}{35} = RS$ - **Answer: Yes** - Expression: $21\tan(R)$ - $\tan(R) = \frac{\text{opposite}}{\text{adjacent}} = \frac{TS}{RS} = \frac{21}{RS}$ - So, $21\tan(R) = 21 \times \frac{21}{RS} = \frac{441}{RS}$, which is not equal to RS. - **Answer: No** **Final answers:** - $35\sin(R)$: No - $21\tan(T)$: Yes - $35\cos(R)$: Yes - $21\tan(R)$: No