Evaluate the value of trigonometry

[harimakenji]

Homework Statement

evaluate
sin (π / 62) . sin (15π/62). sin(23π/62). sin(27π/62) . sin(29π/62)

Homework Equations

trigonometry

The Attempt at a Solution

Tried to change it to cos but failed, tried to use 2 sin A sin B identity and failed again

Thank you very much

 

[Mentallic]

If you use the property that $$sin(x)=cos(\pi/2-x)$$ you can get a nice pattern going. See if that helps.

 

[harimakenji]

sin (π / 62) . sin (15π/62). sin(23π/62). sin(27π/62) . sin(29π/62) = cos (30π / 62). cos(16π/62). cos (8π/62). cos(4π/62). cos(2π/62)

I see no pattern. Is there a nice pattern?

 

[Mentallic]

You don't notice a pattern happening? 2,4,8,16,... now you have 30 instead of 32, but you can change it to 32 with another identity.

 

[harimakenji]

I noticed that but lost because of 30 but now I get it from your clue.

cos (30π / 62) = - cos (32π / 62), then it becomes: - (cos x) . (cos 2x). cos(4x). cos (8x). cos (16x). How to continue that?

 

[vela]

Wikipedia cites a relevant identity:

http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Identities_without_variables

I had never seen it before, though.

 

[Mentallic]

harimakenji, I've done stuff like this in an advanced class in high school (in Australia it is called extension 2 mathematics) but usually they first make us prove the given identity as vela has posted and then finally using the formula to evaluate the question that you've been given. Are you sure this question wasn't a continuation from a bigger problem?

 

[Redbelly98]

Homework Statement

evaluate
sin (π / 62) . sin (15π/62). sin(23π/62). sin(27π/62) . sin(29π/62)

Can you double-check, is the denominator in all those terms supposed to be 62 or 64?

 

[Mentallic]

Can you double-check, is the denominator in all those terms supposed to be 62 or 64?

Actually, it works even more nicely being 62 than 64

 

[harimakenji]

Wikipedia cites a relevant identity:

http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Identities_without_variables

I had never seen it before, though.

I also had never seen it before, phew

harimakenji, I've done stuff like this in an advanced class in high school (in Australia it is called extension 2 mathematics) but usually they first make us prove the given identity as vela has posted and then finally using the formula to evaluate the question that you've been given. Are you sure this question wasn't a continuation from a bigger problem?

The question is just like this, no bigger problem. This is the question for my brother in grade 11.

Can you double-check, is the denominator in all those terms supposed to be 62 or 64?

It is 62. I've checked the result using calculator and it is very nice result, 1 / 32


I can solve this question using the link given by vela. I can't do this without all the help here. Thank you very much for the help (vela, mentallic, redbelly98). I really appreciate all the things I got from here. Once again, thank you.