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

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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) About Me  (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:
    sin%28theta%29+=+cos%2890-theta%29 and
    cos%28theta%29+=+sin%2890-theta%29
  • 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