Question
If α2, b2, c2 are A.P then cot A, cot B, cot C are in
- A.P
- G.P
- H.P
- A.G.P
Hint:
The law of sines says that the ratio of the sine of one angle to the opposite side is the same ratio for all three angles.
The correct answer is: A.P
If α2, b2, c2 are A.P then cot A, cot B, cot C are in
α2, b2, c2 are A.P
It means a2 + c2 = 2b2 .
b2 - a2 = c2 - b2
⇒ k2sin2B − k2sin2A = k2sin2C− k2sin2B using sine rule a=k sinA, b=k sinB and c=k sinC
⇒sin(B+A)sin(B−A)=sin(C+B)sin(C−B)
⇒sin(π−C)sin(B−A)=sin(π−A)sin(C−B) since A+B+C=π
⇒sin C sin(B−A)=sin A sin(C−B)
⇒sin A sin (B−A)=sin C sin(C−B)
⇒sin A sin B sin(B−A)=sin B sin C sin(C−B)
⇒sinA sinB sinB cosA−cosBsinA=sinB sinC sinC cosB−cosB sinC
⇒cotA−cotB=cotB−cotC
∴ cotA, cotB ,cotC are in A.P
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