Solve the following triangle

Question

Answer 1

By Cosine Theorem DE = (34.5² + 27.1² - 2·34.5·27.1cos82.3°)^1/2 = 40.9

By Sine Theorem sinD/27.1 = sin82.3°/40.9; D = sin^-1(27.1sin82.3°/40.9) = 41.0°;

E = 180° - (82.3° + 41.0°) = 56.7°

Answer 2

First, to find the third side f (side opposite angle F), we can use Cosine Law since two sides and included angle are given.

Given:

d=27.1 (Length of EF which is the side opposite <D)

e=34.5 (Length of DF which is the side opposite <E)

m<F=82.3

f²=d²+e²-2(d)(e)Cos(F)

f²=27.1²+34.5²-2(27.1)(34.5)Cos(82.3)

f=40.9

Now that we have length of f, we can use Sine Law to solve for one of the other angles.

Sin(F)/f=Sin(E)/e

Sin(82.3)/40.9=Sin(E)/34.5

Sin(E)=.8359

m<E=sin^-1(.8359)=56.7

Using Triangle Sum Theorem, we can find the third angle.

m<D=180-m<E-m<F=180-56.7-82.3=41