Sin(x+y) and Cos(x+y)

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Sin(x+y) and Cos(x+y)

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Sometimes we need to know the sine or cosine of the sum of two angles (in terms of the sines and cosines of the two angles). Here is a diagram which helps us find out what these are. First, we want to find Sin(x+y). Angles x and y are at point O. Assume that both are acute angles. B is a point on the outside ray of angle y. Draw BA perpendicular (at A) to the ray which is common to the two angles. Draw the other perpendicular lines (AD, BE, and AC) as shown. The second diagram shows how this would look if angles x and y are significantly larger.

  sin(x+y) = BE/OB
           = (BC+CE)/OB
           = (BC+AD)/OB
           = AD/OB + BC/OB
           = (AD/OB)×(OA/OA) + (BC/OB)×(BA/BA)
           = (AD/OA)×(OA/OB) + (BC/BA)×(BA/OB)
  sin(x+y) = sin x cos y + cos x sin y

From this it also follows that sin(2x)=2 sin x cos x. The derivation for the cos(x+y) is similar. Using the same diagram:

  cos(x+y) = OE/OB
           = (OD-DE)/OB
           = (OD-AC)/OB
           = OD/OB - AC/OB
           = (OD/OB)×(OA/OA) - (AC/OB)×(BA/BA)
           = (OD/OA)×(OA/OB) - (AC/BA)×(BA/OB)
  cos(x+y) = cos x cos y - sin x sin y

And it follows that cos(2x)=cos²(x)-sin²(x). There are other forms of this, as cos²(x)+sin²(x)=1 (by the Pythagorean Theorem). And the tan(x+y) can be deduced from the above, and the fact that tan=sin/cos. This comes out to tan(x+y)=(tanx+tany)/(1-tanx tany).

Angles greater than 90 degrees work in these equations. That can be verified in a number of ways.

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