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# 5.11.E: Problems on Exponential and Trigonometric Functions

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Exercise $$\PageIndex{1}$$

Verify formula $$(2)$$.

Exercise $$\PageIndex{2}$$

$$\text { Prove Note } 1, \text { as suggested (using Chapter } 2, §§ 11-12)$$.

Exercise $$\PageIndex{3}$$

Prove formulas $$(1)$$ of Chapter $$2, §§11-12$$ from our new definitions.

Exercise $$\PageIndex{4}$$

Complete the missing details in the proofs of Theorems $$2-4$$.

Exercise $$\PageIndex{5}$$

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 $$x \in E^{1}$$.

Exercise $$\PageIndex{6}$$

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, $$\left.h=q \text { on } E^{1} . \text { Substitute } x=0 \text { to find } q . \text { For (ii), use }(13)-(15) .\right]$$

Exercise $$\PageIndex{7}$$

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
$h(t)=\sin t \cos (p-t)+\cos t \sin (p-t), \quad t \in E^{1} .$
$$\text { Proceed as in Problem } 6 . \text { Then let } t=x .]$$

Exercise $$\PageIndex{8}$$

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 $$\overline{J_{n}}$$.