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The laws of exponents and the algebraic connections between the exponential function and the trigonometric andhyperbolic functions, give the following “addition formulas:”
The following identities hold for all complex numbers $z$ and $w.$
We derive the first formula and leave the others to an exercise.
First, for any two real numbers $x$ and $y,$ we have
which, equating real and imaginary parts, gives that
and
The second of these equations is exactly what we want, but this calculation only shows that it holds for real numbers $x$ and $y.$ We can use the Identity Theorem to show that in fact this formula holds for all complex numbers $z$ and $w.$ Thus, fix a real number $y.$ Let $f\left(z\right)=sinzcosy+coszsiny,$ and let
Then both $f$ and $g$ are power series functions of the variable $z.$ Furthermore, by the previous calculation, $f(1/k)=g(1/k)$ for all positive integers $k.$ Hence, by the Identity Theorem, $f\left(z\right)=g\left(z\right)$ for all complex $z.$ Hence we have the formula we want for all complex numbers $z$ and all real numbers $y.$
To finish the proof, we do the same trick one more time. Fix a complex number $z.$ Let $f\left(w\right)=sinzcosw+coszsinw,$ and let
Again, both $f$ and $g$ are power series functions of the variable $w,$ and they agree on the sequence $\{1/k\}.$ Hence they agree everywhere, and this completes the proof of the first addition formula.
The trigonometric functions $sin$ and $cos$ are periodic with period $2\pi ;$ i.e., $sin(z+2\pi )=sin\left(z\right)$ and $cos(z+2\pi )=cos\left(z\right)$ for all complex numbers $z.$
We have from the preceding exercise that $sin(z+2\pi )=sin\left(z\right)cos\left(2\pi \right)+cos\left(z\right)sin\left(2\pi \right),$ so that the periodicity assertion for the sine function will follow if we show that $cos\left(2\pi \right)=1$ and $sin\left(2\pi \right)=0.$ From part (b) of the preceding exercise, we have that
which shows that $cos\left(2\pi \right)=1.$ Since ${cos}^{2}+{sin}^{2}=1,$ it then follows that $sin\left(2\pi \right)=0.$
The periodicity of the cosine function is proved similarly.
Let $z$ be a nonzero complex number. Prove that there exists a unique real number $0\le \theta <2\pi $ such that $z=r{e}^{i\theta},$ where $r=\left|z\right|.$
HINT: If $z=a+bi,$ then $z=r(\frac{a}{r}+\frac{b}{r}i.$ Observe that $-1\le \frac{a}{r}\le 1,$ $-1\le \frac{b}{r}\le 1,$ and ${\left(\frac{a}{r}\right)}^{2}+{\left(\frac{b}{r}\right)}^{2}=1.$ Show that there exists a unique $0\le \theta <2\pi $ such that $\frac{a}{r}=cos\theta $ and $\frac{b}{r}=sin\theta .$
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