If sinΘ1 + sinΘ2 + sinΘ3 = 3 then cosΘ1 + cosΘ2 + cosΘ3 =?
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...