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Grade: 11
        
find the value  of sin20degree*sin40degree*sin80degree =
10 months ago

Answers : (3)

Lucky
27 Points
							For solving such type of questions you can refer to r.d Sharma of 11 class and chapter 8 (transformation formula) the solution is way too long but I get my answer = square root of 3 divided by 8.
						
10 months ago
Arun
14891 Points
							
I'm assuming that you know this: 
sin(A+B) = sin A cos B + cos A sin B 


Step number 1: 
write sin(40) as sin(60-20) 
write sin(80) as sin(60+20) 
since 60 degrees is a remarkable angle, you know that sin(60) = √(3)/2 

Step 2: 
you have this: 
sin(20)•sin(60-20)•sin(60+20) 

- Simplify sin(60-20)•sin(60+20) 
sin(60-20) = sin60•cos20 - sin20•cos60 
sin(60+20)= sin60•cos20 + sin20•cos60 
- multiply them: 
= sin²60•cos²20 - sin²20•cos²60 

Step 3: 
- you know that sin²A+cos²A=1, 
- so, cos²A = 1 - sin²A 

- Simplify sin²60•cos²20 - sin²20•cos²60 
= sin²60•(1-sin²20) - sin²20•(1-sin²60) 
= sin²60- sin²20•sin²60 - sin²20 + sin²20•sin²60 
= sin²60 - sin²20 
= 3/4 - sin²20 

Step 4: 
This is what we have now: 
= sin20•(3/4 - sin²20) 
= [1/4]•(3sin20 - 4sin³20) 
- note that 4sin³20 comes from putting [(3sin20)/4]-sin³20 on the same base 


Step 5: 
you need to know that 
sin(3A) = 3sin(A) - 4sin³(A) 
(multiple angle relations) 

- so, (3sin20 - 4sin³20) = sin60 

Step 6: 
now you have this: 
= [1/4]•sin60 
= [1/4]•[√(3)/2) 
=√(3)/8
 
 
Regards
Arun (askIITians forum expert)
10 months ago
Avinash
52 Points
							We all know that.   Sinx.sin(60-x)sin(60+x)=1/4sin3xHere x=20Therefore sin20.sin40.sin80=1/4sin3×20                                        =1/4Sin60                                           =1/4×√3/2                                             =√3/8 Sin20.sin40.sin80=√3/8
						
7 months ago
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