5.11.E: Problems on Exponential and Trigonometric Functions
Verify formula \((2)\).
\(\text { Prove Note } 1, \text { as suggested (using Chapter } 2, §§ 11-12)\).
Prove formulas \((1)\) of Chapter \(2, §§11-12\) from our new definitions.
Complete the missing details in the proofs of Theorems \(2-4\).
Prove that
(i) \(\sin 0=\sin (n \pi)=0\);
(ii) \(\cos 0=\cos (2 n \pi)=1\);
(iii) \(\sin \frac{\pi}{2}=1\);
(iv) \(\sin \left(-\frac{\pi}{2}\right)=-1\);
(v) \(\cos \left( \pm \frac{\pi}{2}\right)=0\);
(vi) \(|\sin x| \leq 1\) and \(|\cos x| \leq 1\) for \(x \in E^{1}\).
Prove that
(i) \(\sin (-x)=-\sin x\) and
(ii) \(\cos (-x)=\cos x\) for \(x \in E^{1}\).
[Hint: For (i), let \(h(x)=\sin x+\sin (-x) .\) Show that \(h^{\prime}=0 ;\) hence \(h\) is constant, say, \(\left.h=q \text { on } E^{1} . \text { Substitute } x=0 \text { to find } q . \text { For (ii), use }(13)-(15) .\right]\)
Prove the following for \(x, y \in E^{1} :\)
(i) \(\sin (x+y)=\sin x \cos y+\cos x \sin y ;\) hence \(\sin \left(x+\frac{\pi}{2}\right)=\cos x\).
(ii) \(\cos (x+y)=\cos x \cos y-\sin x \sin y ;\) hence \(\cos \left(x+\frac{\pi}{2}\right)=-\sin x\).
[Hint for \((\mathrm{i}) :\) Fix \(x, y\) and let \(p=x+y .\) Define \(h : E^{1} \rightarrow E^{1}\) by
\[
h(t)=\sin t \cos (p-t)+\cos t \sin (p-t), \quad t \in E^{1} .
\]
\(\text { Proceed as in Problem } 6 . \text { Then let } t=x .]\)
With \(\overline{J_{n}}\) as in the text, show that the sine increases on \(\overline{J_{n}}\) if \(n\) is even and decreases if \(n\) is odd. How about the cosine? Find the endpoints of \(\overline{J_{n}}\).