1) According to the Law of cosines:
3 2 = c 2 + 4 2 − 2 ⋅ 4 c ⋅ c o s 3 7 o 3^2=c^2+4^2-2\sdot4c\sdot cos37^o 3 2 = c 2 + 4 2 − 2 ⋅ 4 c ⋅ cos 3 7 o
c 2 − 8 c ⋅ 0.8 + 16 − 9 = 0 c^2-8c\sdot0.8+16-9=0 c 2 − 8 c ⋅ 0.8 + 16 − 9 = 0
c 2 − 6.4 c + 7 = 0 c^2-6.4c+7=0 c 2 − 6.4 c + 7 = 0
We got a quadratic equation a x 2 + b x + c = 0 ax^2+bx+c=0 a x 2 + b x + c = 0
Its common solution will be
x 1 , 2 = − b ± b 2 − 4 a b 2 a x_1,_2={-b±\sqrt{b^2-4ab}\over2a} x 1 , 2 = 2 a − b ± b 2 − 4 ab
In our case its particular solutions are
c 1 = 6.4 + 6. 4 2 − 4 ⋅ 1 ⋅ 7 2 ⋅ 1 = 6.4 + 12.96 2 = 6.4 + 3.6 2 = 3.2 + 1.8 = 5 c_1={{6.4+\sqrt{6.4^2-4\sdot1\sdot7}}\over{2\sdot1}}={{{6.4+\sqrt{12.96}}\over{2}}={{6.4+3.6}\over{2}}}=3.2+1.8=5 c 1 = 2 ⋅ 1 6.4 + 6. 4 2 − 4 ⋅ 1 ⋅ 7 = 2 6.4 + 12.96 = 2 6.4 + 3.6 = 3.2 + 1.8 = 5
c 2 = 6.4 − 6. 4 2 − 4 ⋅ 1 ⋅ 7 2 ⋅ 1 = 6.4 − 12.96 2 = 6.4 − 3.6 2 = 3.2 − 1.8 = 1.4 c_2={{6.4-\sqrt{6.4^2-4\sdot1\sdot7}}\over{2\sdot1}}={{6.4-\sqrt{12.96}}\over{2}}={{6.4-3.6}\over{2}}=3.2-1.8=1.4 c 2 = 2 ⋅ 1 6.4 − 6. 4 2 − 4 ⋅ 1 ⋅ 7 = 2 6.4 − 12.96 = 2 6.4 − 3.6 = 3.2 − 1.8 = 1.4
2) According to the Law of sines:
s i n A a = s i n B b = s i n C c {sinA\over{a}}={sinB\over{b}}={sinC\over{c}} a s in A = b s in B = c s in C
Now we will find B, C
B 1 = a r c s i n ( b s i n A a ) = a r c s i n ( 4 ⋅ 3 5 3 ) = a r c s i n ( 4 5 ) = 5 3 o B_1={arcsin({{bsinA}\over{a}})}={arcsin({{4\sdot{3\over{5}}}\over{3}})}={arcsin({4\over{5}})}=53^o B 1 = a rcs in ( a b s in A ) = a rcs in ( 3 4 ⋅ 5 3 ) = a rcs in ( 5 4 ) = 5 3 o
C 1 = a r c s i n ( c s i n A a ) = a r c s i n ( 5 ⋅ 3 5 3 ) = a r c s i n ( 1 ) = 9 0 o C_1={arcsin({{csinA}\over{a}})}={arcsin({{5\sdot{3\over{5}}}\over{3}})}={arcsin(1)}=90^o C 1 = a rcs in ( a cs in A ) = a rcs in ( 3 5 ⋅ 5 3 ) = a rcs in ( 1 ) = 9 0 o
B 2 = a r c s i n ( b s i n A a ) = a r c s i n ( 4 ⋅ 3 5 3 ) = a r c s i n ( 4 5 ) = 12 7 o B_2={arcsin({{bsinA}\over{a}})}={arcsin({{4\sdot{3\over{5}}}\over{3}})}={arcsin({4\over{5}})}=127^o B 2 = a rcs in ( a b s in A ) = a rcs in ( 3 4 ⋅ 5 3 ) = a rcs in ( 5 4 ) = 12 7 o
C 2 = a r c s i n ( b s i n A a ) = a r c s i n ( 1 , 4 ⋅ 3 5 3 ) = a r c s i n ( 7 25 ) = 1 6 o C_2={arcsin({{bsinA}\over{a}})}={arcsin({{1,4\sdot{3\over{5}}}\over{3}})}={arcsin({7\over{25}})}=16^o C 2 = a rcs in ( a b s in A ) = a rcs in ( 3 1 , 4 ⋅ 5 3 ) = a rcs in ( 25 7 ) = 1 6 o
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