Question

solve the right triangle.

write your answers as integers or as decimals rounded to the nearest tenth.
$st = $
$rs = $
$m\angle t = \quad ^\circ$
submit
work it out

Explanation:

Step1: Find $\angle T$

The sum of angles in a triangle is $180^\circ$. For right triangle $RST$, $\angle R=90^\circ$, $\angle S=40^\circ$.
$\angle T = 180^\circ - 90^\circ - 40^\circ = 50^\circ$

Step2: Calculate $RS$

Use tangent of $\angle S$: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan(40^\circ)=\frac{RT}{RS} \implies RS=\frac{RT}{\tan(40^\circ)}$
Substitute $RT=5.9$, $\tan(40^\circ)\approx0.8391$:
$RS=\frac{5.9}{0.8391}\approx6.9$

Step3: Calculate $ST$

Use cosine of $\angle S$: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.
$\cos(40^\circ)=\frac{RS}{ST} \implies ST=\frac{RS}{\cos(40^\circ)}$
Substitute $RS\approx6.9$, $\cos(40^\circ)\approx0.7660$:
$ST=\frac{6.9}{0.7660}\approx9.1$
(Alternatively, use Pythagorean theorem: $ST=\sqrt{RT^2+RS^2}=\sqrt{5.9^2+6.9^2}\approx9.1$ for verification)

Answer:

$ST = 9.1$
$RS = 6.9$
$m\angle T = 50^\circ$