What is the solution to finding the sum of square sin() functions?

[Aikon]

Hi all,
I work as monitor for a pre-university course and one student showed me this problem:

Homework Statement

To find how much is y:
y=sin²(10)+sin²(20)+sin²(30)+...+sin²(80)+sin²(90)

Homework Equations

I don't know. I thought about sen²x+cos²x=1

The Attempt at a Solution

To use the equation given above and to write a series of cos()'s, but i don't think it get better than before.

 

[ehild]

Homework Equations

I thought about sin²x+cos²x=1

You are on the right track. Use also that sin(x)=cos(90-x). For example, sin(80°)=cos(10°).

ehild

 

[Mark44]

There's another identity that I think will be helpful:
$$sin^2(x) = \frac{1 - cos(2x)}{2}$$

Also, cos(x) = -cos($\pi$ - x)

 

[Aikon]

Thank you all the answers.

I liked this:

You are on the right track. Use also that sin(x)=cos(90-x). For example, sin(80°)=cos(10°).

ehild

With this identity it becomes almost trivial, because (sin²10 +sin²80)=1 and it goes like this for other pairs, it gives 5 in the end.