It is very useful to know how to find the sine and cosine of combined angles, as well as how to find the sum of sines and cosines for two angles. These are useful in trigonometry generally, but especially when trying to prove trigonometric identities or solve trigonometric equations.
sin(x+y) and cos(x+y), sin(x-y) and cos(x-y)
The formula for the sines and cosines of added and subtracted angles are as follows:
sin(x+y) = sin(x).cos(y) + sin(y).cos(x) ... Equation 1
sin(x-y) = sin(x).cos(y) - sin(y).cos(x) ... Equation 2
cos(x+y) = cos(x).cos(y) - sin(x).sin(y) ... Equation 3
cos(x-y) = cos(x).cos(y) + sin(x).sin(y) ... Equation 4
The proofs for equations 1 & 3 are described in "Sin(a+b) Proof Using De Moivre's Theorem - Sine Sum of Angles"
Since cos() is an even function, and sin() is an odd function, then
cos(-A) = cos(A) (Equation 5) and sin(-A) = -sin(A) (Equation 6).
So, given that Equation 1 is true:
sin(x-y) = sin(x).cos(-y) + sin(-y).cos(x), but from equation 5 cos(-y) = cos(y) and sin(-y) = -sin(y)
= sin(x).cos(y) - sin(y).cos(x) which proves equation 2.
Similarly, starting with equation 3
cos(x+y) = cos(x).cos(y) - sin(x).sin(y), so
cos(x-y) = cos(x).cos(-y) - sin(x).sin(-y)
= cos(x).cos(y) + sin(x).sin(y) which proves equation 4.
Trigonometric Formula sin(a)+sin(b) and cos(a)+cos(b)
Equation 1 + Equation 2:
sin(x+y) + sin(x-y) = sin(x).cos(y) + sin(y).cos(x) + sin(x).cos(y) - sin(y).cos(x)
= 2.sin(x).cos(y) ... Equation 7.
Now, introduce two variables P and Q such that
P = (x+y) and
Q= (x-y)
So x = (P + Q) / 2 and
y = (P - Q) / 2
Substituting these into equation 7:
sin(P) + sin(Q) = 2.sin( (P+Q) / 2).cos( (P - Q) / 2)
A similar method is used to find cos(A) + cos(B):
cos(x+y) = cos(x).cos(y) - sin(x).sin(y) ... equation 3
cos(x-y) = cos(x).cos(y) + sin(x).sin(y) ... equation 4.
Adding equations 3 and 4 gives:
cos(x+y) + cos(x-y) = cos(x).cos(y) - sin(x).sin(y) + cos(x).cos(y) + sin(x).sin(y)
= 2.cos(x).cos(y) ... equation 9
Now, introduce two variables P and Q such that
P = (x+y) and
Q= (x-y)
So x = (P + Q) / 2 and
y = (P - Q) / 2
Substituting into equation 9 gives:
cos(P) + cos(Q) = 2.cos( (P+Q) / 2).cos( (P-Q) / 2)
So the formula for the sum of the sines of two angles and the cosines of two angles have been derived.
Sin(a+b), Cos(a+b), Sin(a)+Sin(b), Cos(a)+Cos(b) Summary
Manipulating trigonometric functions is often important in math, science and engineering. The trigonometric identities for the sum of two angles, and the sum of sines or cosines, are commonly used and useful to know. The identities have been stated here. The proofs have not been rigorously derived, but have been sketched out in a way that tries to make them easily understood and easily remembered. Trigonometric algebra is a key element of most math high school and degree courses.
Sin(a+b), Cos(a+b), Sin(a)+Sin(b), Cos(a)+Cos(b) References
The articles "Trigonometric Identities Lesson" and "Trigonometric Identity Advanced Example" describe why and how to manipulate trigonometric identities and functions, and may be read along with this article so as to expand the reader's understanding of general trigonometry. The site jimloy.com shows the geometric derivation of the formula for the function sin(a+b).
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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