Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

5.11.E: Problems on Exponential and Trigonometric Functions

  • Page ID
    24107
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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 \(\overline{J_{n}}\).