Question

22. Prove that, sin20°sin40°sin80° = √3/8​

Answer 1

Step-by-step explanation:

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

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Answer 2

Step-by-step explanation:

sin20°sin40°sin80°

(sinA. sin(60-A) . sin(60+A) = sin3A/4

similarly using this formula

sin20°sin40°sin80° = sin 3×20/4

= √3/2/4

= √3/8

So, we proved that ,

sin20°sin40°sin80° = √3/8

Hope it helps you buddy