Question

solve the right triangle.
t
3.8
s
32°
r
write your answers as integers or as decimals rounded to the nearest tenth.
rt =
rs =
m∠t = °

Explanation:

Step1: Find \( m\angle T \)

In a right triangle, the sum of the acute angles is \( 90^\circ \). Given \( m\angle R = 32^\circ \) and \( \angle S = 90^\circ \), we have:
\( m\angle T = 90^\circ - 32^\circ = 58^\circ \)

Step2: Find \( RT \) (hypotenuse)

We know the opposite side to \( \angle R \) is \( TS = 3.8 \). Using the sine function: \( \sin(R) = \frac{opposite}{hypotenuse} \), so \( \sin(32^\circ) = \frac{3.8}{RT} \).
Solving for \( RT \): \( RT = \frac{3.8}{\sin(32^\circ)} \).
\( \sin(32^\circ) \approx 0.5299 \), so \( RT \approx \frac{3.8}{0.5299} \approx 7.2 \)

Step3: Find \( RS \) (adjacent side to \( \angle R \))

Using the tangent function: \( \tan(R) = \frac{opposite}{adjacent} \), so \( \tan(32^\circ) = \frac{3.8}{RS} \).
Solving for \( RS \): \( RS = \frac{3.8}{\tan(32^\circ)} \).
\( \tan(32^\circ) \approx 0.6249 \), so \( RS \approx \frac{3.8}{0.6249} \approx 6.1 \)

Answer:

\( RT \approx 7.2 \)
\( RS \approx 6.1 \)
\( m\angle T = 58^\circ \)