SOLUTION: If the Expression, 2cos10 + sin100 + sin1000 + sin10000 where all angles are in degrees then the whole expression simplifies to,,?

Question 721675: If the Expression, 2cos10 + sin100 + sin1000 + sin10000 where all angles are in degrees then the whole expression simplifies to,,?
Answer by jsmallt9(3759) (Show Source): You can put this solution on YOUR website!
There are two keys to this problem are:

With complementary angles, the sin of one is the cos of the other (and vice versa). In formula form:
and
sin's of co-terminal angles are equal. And co-terminal angles are multiples of 360 degrees apart.

First let's establish a fact we will use several times. 80 and 10 degrees are complementary (because they add up to 90). So:
sin(80) = cos(10)

Now let's look at the sin terms:

sin(100)
100 degrees terminates in the second quadrant with a reference angle of 80 degrees. And since sin's in the second quadrant are positive:
sin(100) = sin(80) = cos(10)
sin(1000)
1000 degrees is 80 degrees short of 3*360. So 1000 degrees will terminate in the 4th quadrant with a reference angle of 80 degrees. sin is negative in the 4th quadrant so:
sin(1000) = -sin(80) = -cos(10)
sin(10000)
10000 degrees is 80 degrees short of 27*360. So it is co-terminal with sin(1000) and will have the same sin:
sin(10000) = -sin(80) = -cos(10)

Now let's put all this together. Replacing all the sin's in
2cos(10) + sin(100) + sin(1000) + sin(10000)
with the cos expressions we found for them we get:
2cos(10) + cos(10) + (-cos(10)) + (-cos(10))
These are all like terms now so we can add them:
cos(10) is the simplified expression
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