Question

In ΔABC, ∠C is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth.

Answer 1

Answer:

The remaining side and angles are $a = 20\sqrt{3}$, $\angle A = 60^{\circ}$ and $\angle B = 30^{\circ}$.

Step-by-step explanation:

According to the information given on statement, we are in front of a right triangle because $\angle C$, opossite to side $c$, is a right angle. Hence, $c$ is the hypotenuse of the right triangle and $b$, a leg. The missing length can be calculated by the Pythagorean Theorem:

$a = \sqrt{c^{2}-b^{2}}$ (1)

If we know that $c = 40$ and $b = 20$, then the length of the missing leg is:

$a = \sqrt{c^{2}-b^{2}}$

$a = 20\sqrt{3}$

Lastly, we find the value of the missing angles by means of direct and inverse trigonometric relations:

Angle A

$\angle A = \tan^{-1}\left(\frac{a}{b} \right)$ (2)

Angle B

$\angle B = \tan^{-1} \left(\frac{b}{a} \right)$ (3)

If we know that $a = 20\sqrt{3}$ and $b = 20$, then the values of the missing angles are, respectively:

Angle A

$\angle A = \tan^{-1}\left(\frac{a}{b} \right)$

$\angle A = 60^{\circ}$

Angle B

$\angle B = \tan^{-1} \left(\frac{b}{a} \right)$

$\angle B = 30^{\circ}$

The remaining side and angles are $a = 20\sqrt{3}$, $\angle A = 60^{\circ}$ and $\angle B = 30^{\circ}$.