Solve the following triangles

Question

1. Recall that "to solve a triangle" means to find values for all the angles and sides that weren't already given values. Using your knowledge of the sine and cosine laws, solve the following triangles. (5 marks each)
2. Triangle ABC in which ∠A=65°, ∠B=35°, and AB=7 units.
3. Triangle XYZ in which x=8.0 cm, y=7.0 cm, and z=5.5 cm.
4. Triangle DEF in which ∠D=73°, d=14 cm, and f=10 cm.
5. Triangle KMN in which k=90 cm, n=80 cm, and ∠M=50°.

Answer 1

angle C = 180-65+35 = 80 degrees

AB/sin80 = BC/sin65 = AC/sin35

7/.985 = BC/.906

BC = .7(.906)/.985 = about 6.44

AC = 7(.574)/.985 = about 4.08

5.5^2 = 8^2 +7^2 - 2(8)(7)cosZ

30.25 = 64+49 -112cosZ

cosZ = (113-30.25)/112 =.739

Z = cos^-1(.739)= about 42.37 degrees

x/sinX = z/sinZ

8/sinX = 5.5/sin42.37

sinX = 8sin42.37/5.5= .980

X = sin^-1(.980)=about 78.58 degrees

Y = 180-42.37-78.58 = 59.05 degrees

the largest side is opposite the largest angle

smallest side is opposite the smallest angle

Law of Sines

sin73/14 = sinF/10

sinF = 10sin73/14 = .683

F= 43.084 degrees

Law of Cosines

m^2 =90^2+80^2 -2(90)(80)cos50

= 8100+6400-14400(.642876)

= 14500- 9256.1416

= 5243.858421

m=sqr(5243.858421)

m=72.41449

m= about 72 cm