solve the triangles

Question

1) c = 26 , <B = 49 , <c = 53

2) <A = 56 , b = 12 , a = 11

3) <A= 94 , <C = 22 , b=5

4) <C = 58, a = 12, b = 16

5) <B = 68 , b = 14, c = 18

Answer 1

I will solve the first one. See if you can follow me  and solve the others.

c = 26 , <B = 49° , <C = 53°

First of all, for any triangle, <A + <B + <C = 180°.

Therefore <A = 180° - 49° - 53° = 78°

Now use the law of sines: a/sin A = b/sin B = c/sin C.

You get a/sin 78° = b/sin 49° = 26/sin 53°

Solving for a and b gives

a = 26(sin 78°) / sin 53° = 31.84

b = 26(sin 49°) / sin 53° = 24.56

Answer 2

1) (sin 53)/26 = (sin 49)/b
b*sin 53 = 26*sin 49
b = (26*sin 49)/(sin 53)
b = 24.6

A = 180-(53+49)
A = 78

(sin 53)/26 = (sin 78)/a
a = (26*sin 78)/(sin 53)
a = 31.8


2) (sin 56)/11 = (sin B)/12
sin B = (12*sin 56)/11
B = arcsin((12* sin56)/11)
B = 64.74
C = 180-(56+64.74)
C = 59.26

(sin 56)/11 = (sin 59.29)/c
c*sin 56 = 11*sin 59.26
c = (11*sin 59.26)/(sin 56)
c = 11.4


3) B = 180-(94+22)
B = 64

(sin 64)/5 = (sin 94)/a
a*sin 64 = 5*sin 94
a = (5*sin 94)/(sin 64)
a = 5.5

(sin 64)/5 = (sin 22)/c
c*sin 64 = 5*sin 22
c = (5*sin 22)/(sin 64)
c = 2.1


4) c^2 = a^2 + b^2 - 2ab*cosC
c^2 = 144+256-384*cos 58
c^2 = 196.5
c = 14

(sin 58)/14 = (sin A)/12
14*sin A = 12*sin 58
sin A = (12*sin 58)/14
A = arcsin((12*sin 58)/14)
A = 46.6

B = 180-(46.6+58)
B = 75.4


5) side b is not long enough to create a triangle

Answer 3

The law of sines can help with this: (a/sin<A)=(b/sin<B)=(c/sin<C)

1) <A = 180-102= 78, (a/sin78)=(26/sin53) => a≈31.8,  (b/sin49)=(26/sin53) =>  b≈24.5

2) (11/sin56)=(12/sin<B) =>  <B≈64.8,  <C=59.2,  (11/sin56)=(c/sin59.2) =>  c≈11.4

3) <B=64,  (a/sin94)=(5/sin64) =>  a≈5.55,  (c/sin22)=(5/sin64) =>  c≈2.1

4) Law of Cosine: c²= a² + b² - 2ab(cos<C),  c² = 144+256-384cos58 =>  c=14,  (12/sin<A)=(14/sin58) =>  <A≈46.5,  (16/sin<B)=(14/sin58) =>  <B≈75.5

5) 196= a²+324-36a(cos68) => 0=a² -13.5a +128 (complete the square) => -82.4=(a-6.75)² =>  a≈2.3,   (14/sin68)=(2.3/sin<A) =>  <A≈8.8,  180-8.8-68 =>  <C=103.2

Hope this helps!