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If any triangle ABC, find the value of a...

If any triangle ABC, find the value of `asin(B-C)+b sin(C-A)+c sin(A-B)dot`

Text Solution

Verified by Experts

Use sin formula
`sinA/a​=sinB/b​=sinC/c​=k`
`sinA=ak,sinB=bk,sinc=ck`
Also,
`sin(A−B)=sinA..cosB−cosA−sinB`
=`akcosB−cosA.bk`
=`k(acosB−bcosA)`
Similarity,
`sin(B−C)=k(bcosC−ccosB)`
`sin(C−A)=k(c cosA−acosc)`
...
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RD SHARMA-SINE AND COSINE FORMULAE AND THEIR APPLICATIONS-Solved Examples And Exercises
  1. In a DeltaA B C , prove the following : c\ (acosB-bcosA)=a^2-b^2dot

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  2. In a DeltaA B C , prove the following : (c^2+b^2-a^2)tanA=(a^2+c^2-b^...

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  3. In a Delta A B C , prove the following : (c-b\ cos A)/(b-c\ cos A)=(c...

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  4. In a Delta A B C , prove the following : \ a(cos B+cos C-1)+b(cos C+c...

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  5. In a DeltaA B C , prove the following : a\ cos A+b\ cos B+c cos C=2as...

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  6. In a Delta A B C , prove the following : a^2=(b+c)^2-4bc cos^2(A/2)dot

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  7. In a DeltaA B C , prove the following : 4\ (b c cos^2(A/2)+c acos^2(B...

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  8. In a DeltaA B C ,\ prove that sin^3Acos(B-C)+sin^3B cos(C-A)+sin^3\ ...

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  9. In any DeltaA B C ,(b+c)/(12)=(c+a)/(13)=(a+b)/(15),\ then prove tha...

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  10. If in a Delta A B C ,cos^2A+cos^2B+cos^2C=1, prove that the triangle ...

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  11. Two ships leave a port at the same time. One goes 24 km/hr in the d...

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  12. Find the area of the triangle DeltaA B C in which a=1,\ b=2\ a n d\ /...

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  13. In a DeltaA B C ,\ if\ b=sqrt(3),\ c=1\ a n d\ /A=30^0, then find a

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  14. In a Delta A B C , if cos A=( sin B)/(2sin C), then show that c=adot

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  15. In a DeltaA B C ,\ ifb=20 ,\ c=21\ a n dsinA=3/5 find adot

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  16. In DeltaA B C ,\ if\ a=8,\ b=10 ,\ c=12\ a n d\ C=lambdaA ,\ find th...

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  17. If the sides of a triangle are proportional to 2,\ sqrt(6)\ a n d\ sqr...

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  18. If in a "Delta"A B C ,(cosA)/a=(cosB)/b=(cosC)/c , then find the measu...

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  19. If any triangle ABC, find the value of asin(B-C)+b sin(C-A)+c sin(A-B)...

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  20. In any DeltaABC , find the value of suma(sinB-sin C)dot

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