Considering L.H.S : Getting terms c – b cos A and b – c cos A from projection formula Using projection formula, \[c\text{ }=\text{ }a\text{ }cos\text{ }B\text{ }+\text{ }b\text{ }cos\text{...
For any Δ ABC show that (c2 – a2 + b2) tan A = (a2 – b2 + c2) tan B = (b2 – c2 + a2) tan C
Considering L.H.S : \[({{c}^{2}}~\text{ }{{a}^{2}}~+\text{ }{{b}^{2}}),\text{ }({{a}^{2}}~\text{ }{{b}^{2}}~+\text{ }{{c}^{2}}),\text{ }({{b}^{2}}~\text{ }{{c}^{2}}~+\text{ }{{a}^{2}})\] Using sine...
For any Δ ABC show that 2 (bc cos A + ca cos B + ab cos C) = a2 + b2 + c2
Considering L.H.S : \[2\text{ }\left( bc\text{ }cos\text{ }A\text{ }+\text{ }ca\text{ }cos\text{ }B\text{ }+\text{ }ab\text{ }cos\text{ }C \right)\] \[2ca\text{ }cos\text{ }B,\text{ }2ab\text{...
For any Δ ABC show that c (a cos B – b cos A) = a2 – b2
Considering L.H.S. \[c\text{ }\left( a\text{ }cos\text{ }B\text{ }\text{ }b\text{ }cos\text{ }A \right)\] \[ca\text{ }cos\text{ }B\text{ }and\text{ }cb\text{ }cos\text{ }A\] is present in L.H.S,...
For any ΔABC, show that b (c cos A – a cos C) = c2 – a2
So, considering LHS: \[b\text{ }\left( c\text{ }cos\text{ }A\text{ }\text{ }a\text{ }cos\text{ }C \right)\] \[bc\text{ }cos\text{ }A\text{ }and\text{ }ab\text{ }cos\text{ }C\]are L.H.S, which can be...
In a ∆ABC, if a = 18, b = 24, c = 30, find cos A, cos B and cos C
ACCORDING TO QUES,: Sides of a triangle are \[a\text{ }=\text{ }18,\text{ }b\text{ }=\text{ }24\text{ }and\text{ }c\text{ }=\text{ }30\] Hence, using the formulas: \[Cos\text{ }A\text{ }=\text{...
The sides of a triangle are a = 4, b = 6 and c = 8, show that: 8 cos A + 16 cos B + 4 cos C = 17.
ACCORDING TO QUES,: Sides of the triangle: \[a\text{ }=\text{ }4,\text{ }b\text{ }=\text{ }6\text{ }and\text{ }c\text{ }=\text{ }8\] By using the formulas, \[Cos\text{ }A\text{ }=\text{...
In a ∆ABC, if a = √2, b = √3 and c = √5 show that its area is1/2 √6 sq. units.
According to the given ques: In \[\vartriangle ABC,\text{ }a\text{ }=\text{ }\surd 2,\text{ }b\text{ }=\text{ }\surd 3\text{ }and\text{ }c\text{ }=\text{ }\surd 5~\] Hence, using the formulas,...
In a ∆ABC, if a = 5, b = 6 and C = 60o, show that its area is (15√3)/2 sq. units.
ACCORDING TO QUES: In a \[\vartriangle ABC,\text{ }a\text{ }=\text{ }5,\text{ }b\text{ }=\text{ }6\text{ }and\text{ }C\text{ }=\text{ }{{60}^{o}}\] By using the formula, Area of\[\vartriangle...
prove the following: a^2sin(B-C)/sin A + b^2sin(C-A)/sin B+ c^2sin(A-B)/sin C= 0
on simplification we have, \[\begin{array}{*{35}{l}} =\text{ }{{k}^{2}}~[sin\text{ }A\text{ }sin\text{ }\left( B\text{ }\text{ }C \right)\text{ }+\text{ }sin\text{ }B \\ sin\text{ }\left( C\text{...
Prove the following: a (sin B – sin C) + b (sin C – sin A) + c (sin A – sin B) = 0
according to sine rule, \[a\text{ }=\text{ }k\text{ }sin\text{ }A,\] \[~b\text{ }=\text{ }k\text{ }sin\text{ }B,\] \[c\text{ }=\text{ }k\text{ }sin\text{ }C\] Let LHS: \[a\text{ }\left( sin\text{...
prove the following:$\sqrt{\sin A}-\sqrt{\sin B}$ /$\sqrt{\sin A}+\sqrt{\sin B}$ $=a+b-2\sqrt{ab}/a-b$ .
according to the ques, now, equals RHS, hence proved.
Prove the following: a2 sin (B – C) = (b2 – c2) sin A
According to sine rule, now, \[~c\text{ }=\text{ }k\text{ }sin\text{ }C\] same way, \[a\text{ }=\text{ }k\text{ }sin\text{ }A\] also, \[b\text{ }=\text{ }k\text{ }sin\text{ }B\] As we know, Now let...
Prove the following: b sin B – c sin C = a sin (B – C)
according to sine rule, Now, \[~c\text{ }=\text{ }k\text{ }sin\text{ }C\] In the same way, \[a\text{ }=\text{ }k\text{ }sin\text{ }A\] also, \[~b\text{ }=\text{ }k\text{ }sin\text{ }B\] As we know,...
Prove the following: a^2-c^2/b^2=sin(A-C)/sin(A+C)
According to the given equation we can use sine rule, now,
Prove the following: sin(B-C/2)=b-c/a (cosA/2)
according to sine rule, now,
Prove the following : a+b/c=cos(A-B/2)/sinC/2
According to sine rule,
prove the following: c/a+b=1-tan(A/2)tan(B/2)/1+tan(A/2)tan(B/2)
according to sine rule, \[cos\text{ }\left( A\text{ }+\text{ }B \right)/2\text{ }=\text{ }cos\text{ }\left( A/2\text{ }+\text{ }B/2 \right)\text{ }=\text{ }cos\text{ }A/2\text{ }cos\text{ }B/2\text{...
Prove the following : c/a-b= tan(A/2)+tan(B/2)/ tan(A/2)-tan(B/2)
according to sine rule,
Prove the following: (a – b) cos C/2 = C sin (A – B)/2
according to sine rule, equals RHS. Hence proved.
In any triangle ABC, prove the following: a-b/a+b=tan(A-B/2)/tan(A+B/2)
according to sine rule, equals RHS. hence proved.
In ∆ABC, if a = 18, b = 24 and c = 30 and ∠C = 90o, find sin A, sin B and sin C.
according to sine rule,
If in any ∆ABC, ∠C = 105o, ∠B = 45o, a = 2, then find b.
According to ques, : In ∆ABC, \[\angle C\text{ }=\text{ }{{105}^{o}},\] \[~\angle B\text{ }=\text{ }{{45}^{o}},\] \[~a\text{ }=\text{ }2\] Since, \[\angle A\text{ }+~\angle B\text{ }+~\angle...
If in a ∆ABC, ∠A = 45o, ∠B = 60o, and ∠C = 75o; find the ratio of its sides.
according to sine rule, \[a:\text{ }b:\text{ }c\text{ }=\text{ }2:\text{ }\surd 6:\text{ }\left( 1+\surd 3 \right)\] Therefore, the ratio of the sides of the given triangle is: \[~a:\text{ }b:\text{...