If sinΘ_{1} + sinΘ_{2} + sinΘ_{3} = 3 then cosΘ_{1} + cosΘ_{2} + cosΘ_{3 }=?
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2 Answers
As mentioned, the maximum value of the sine function is 1. Thus, the only way the sum of the sines of 3 distinct angles can be 3 is if each sine is equal to 1. The only angles for which sine is 1 (in radians) are ∏/2+2N∏, where N covers all natural numbers
(note I am using capital pi here to represent 3.1415... or 180 deg... b/c lowercase pi looked like an "n")...in other words 90 degrees or any other
version, so to say, of 90 degrees (i.e. any multiple of 360 degrees added to 90 deg) has its sine as 1.
By the Pythagorean identity, the sum of the squares of sine and cosine of the same angle is always 1. Hence, if sine of an angle is already 1, what must the cosine of that angle be?
This should answer the question...
The maximum the sine can be is 1. Therefore, each theta has to be either at 0, pi, 2pi, etc. (0, 18, 360, ...)
You should look at a picture of the sine graph and the cosine graph. The cosine is the same shape as the sine but off by 90 degrees or pi/2. When the sine is 1, the cosine is 0. You can also see this on your unit circle diagram which you should review.
Therefore you can see what the sum of the cosines of those angles will be.
Dorene O.