The article "Trigonometry Sin(a+b) Cos(a+b) Sin(a)+Sin(b) Cos(a)+Cos(b)" shows how to derive several trigonometric identities, but it assumes that the formula for sin(x+y) is already known. Several websites describe the proof of the formula for sin(x+y) using trigonometry, but this article uses a simpler method based on De Moivre's Theorem. It is assumed that the student is familiar with the basic rules of logarithms and exponents.
Sin(x+y) Proof - Background
There are many instances where there is a need to know, or prove, the formula for sin(x+y). This is true for the later years of High School mathematics, as well as college math. The formula is a precursor to several other trigonometric proofs. The following identities are assumed, although they are proven elsewhere:
De Moivre's Theorem: (cosx + i.sinx)^n = cos(nx) + i.sin(nx)
where i = √(-1) by convention. (In some literature "j" is used instead of "i", so as not to confuse it with the symbol for electrical current.)
This is proven in the article "De Moivres Theorem Description With Examples and Application"
Euler's Formula: e^(ix) = cosx + i.sinx,
where e = base of natural logarithms, 2.71828...
Law of Exponents: a^(x+y) = a^x × a^y,
which is proven in the article "Basic Logarithms And Exponents Explanation And Lesson Plan"
Sin(x+y) Proof
It is known from Euler's formula that
e^(i.x) = cos(x) + i.sin(x)
Letting x = a + b,
e^(i.x) = e^(i.(a+b))
= e^(i.a + i.b)
= e^(i.a) × e^(i.b), from the Law of Exponents
But e^(i.(a+b)) = cos(a+b) + i.sin(a+b) from Euler's formula, so
e^(i.a) × e^(i.b) = cos(a+b) + i.sin(a+b)
Expanding the left-hand side,
[cos(a) + i.sin(a)] × [cos(b) + i.sin(b)] = cos(a+b) + i.sin(a+b), so
cos(a).cos(b) + i².sin(a).sin(b) + i.sin(a).cos(b) + i. cos(a).sin(b) = cos(a+b) + i.sin(a+b), so
cos(a).cos(b) - sin(a).sin(b) + i.(sin(a).cos(b) + cos(a).sin(b)) = cos(a+b) + i.sin(a+b)
Equating real and imaginary parts,
cos(a).cos(b) - sin(a).sin(b) = cos(a+b), and
sin(a).cos(b) + cos(a).sin(b) = sin(a+b)
Q.E.D.
So the proofs for the sum of sines and cosines of angles have been shown very simply.
Summary of Sin(x+y) and Cos(x+y) Uses
Many other trigonometric proofs depend on the formula for sin(x+y) and cos(x+y). The result is widely used throughout math courses at many levels, and the result is used throughout engineering, including electronics and construction. The proof shown here is shorter and more easily understood than the more widely-known proof that uses two right-angled triangles.
References For Proof of Sin(a+b) and Cos(a+b)
The proofs shown here depend only on the Law of Exponents, Euler's formula, and De Moivre's Theorem, and no other references have been used. (De Moivre's Theorem is not really a fundamental proof either, since it is derived from Euler's formula). One additional exercise that is useful for the student is to try to prove the formula for Sin(a+b+c): this should be tried using the trigonometric method, and then using the method used here based on Euler's formula. That way, the real power of Euler's formula will be seen.
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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