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Question

In a ABC, if sin3A=0, then it is

A
equilateral
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B
right angled
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C
isosceles
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D
has at least one angle 600
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Solution

The correct options are
A equilateral
D has at least one angle 600
sin3A=0
sin3A+sin3B+sin3C=0 by expanding the summation
2sin(32(A+B))cos(32(AB))+sin3C=0 using compound angle formula.
2sin(32(A+B))cos(32(AB))+2sin3C2cos3C2=0 using multiple angle formula.
2sin(3π23C2)cos(32(AB))+2sin3C2cos3C2=0 since A+B+C=π
2cos3C2cos(32(AB))+2sin3C2cos3C2=0
cos3C2[cos(32(AB))cos3C2]=0 by taking common terms
cos3C2[cos(32(AB))+cos(32(A+B))]=0 since cos(πθ)=cosθ
cos3C2cos3A2cos3B2=0 by compound angle formula
cos3C2=cos900
3C2=900
C=600
C=600 or A=600 or B=600

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