Question

solve the right triangle. round decimal answers to the nearest tenth.
rs≈□, rt≈□, m∠t=□°

Explanation:

Step1: Find $m\angle T$

In a right triangle, acute angles sum to $90^\circ$.
$m\angle T = 90^\circ - 57^\circ = 33^\circ$

Step2: Calculate $RS$

Use tangent of $\angle R$: $\tan(57^\circ)=\frac{TS}{RS}$
$RS = \frac{TS}{\tan(57^\circ)} = \frac{15}{\tan(57^\circ)} \approx \frac{15}{1.5399} \approx 9.7$

Step3: Calculate $RT$

Use cosine of $\angle R$: $\cos(57^\circ)=\frac{RS}{RT}$
$RT = \frac{RS}{\cos(57^\circ)} \approx \frac{9.7}{\cos(57^\circ)} \approx \frac{9.7}{0.5446} \approx 17.8$
(Alternatively, use Pythagorean theorem: $RT=\sqrt{RS^2 + TS^2}=\sqrt{9.7^2+15^2}\approx17.8$)

Answer:

$RS \approx 9.7$, $RT \approx 17.8$, $m\angle T=33^\circ$