7.1E: Exercises
- Page ID
- 19011
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Verbal
1) We know \(g(x)=\cos x\) is an even function, and \(f(x)=\sin x\) and \(h(x)=\tan x\)are odd functions. What about \(G(x)=\cos ^2 x\), \(F(x)=\sin ^2 x\) and \(H(x)=\tan ^2 x\)? Are they even, odd, or neither? Why?
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All three functions, \(F,G,\) and \(H\) are even.
This is because
\(F(-x)=\sin(-x)\sin(-x)=(-\sin x)(-\sin x)=\sin^2 x=F(x),G(-x)=\cos(-x)\cos(-x)=\cos x\cos x= cos^2 x=H(-x)=\tan(-x)\tan(-x)=(-\tan x)(-\tan x)=\tan2x=H(x)\)
2) Examine the graph of \(f(x)=\sec x\) on the interval \([-\pi ,\pi ]\)How can we tell whether the function is even or odd by only observing the graph of \(f(x)=\sec x\)?
3) After examining the reciprocal identity for \(\sec t\) explain why the function is undefined at certain points.
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When \(\cos t = 0\) then \(\sec t = 10\) which is undefined.
4) All of the Pythagorean identities are related. Describe how to manipulate the equations to get from \(\sin^2t+\cos^2t=1\) to the other forms.
Algebraic
For the exercises 5-15, use the fundamental identities to fully simplify the expression.
5) \(\sin x \cos x \sec x\)
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\(\sin x\)
6) \(\sin(-x)\cos(-x)\csc(-x)\)
7) \(\tan x\sin x+\sec x\cos^2x\)
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\(\sec x\)
8) \(\csc x+\cos x\cot(-x)\)
9) \(\dfrac{\cot t+\tan t}{\sec (-t)}\)
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\(\csc x\)
10) \(3\sin^3 t\csc t+\cos^2 t+2\cos(-t)\cos t\)
11) \(-\tan(-x)\cot(-x)\)
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\(-1\)
12) \(\dfrac{-\sin (-x)\cos x\sec x\csc x\tan x}{\cot x}\)
13) \(\dfrac{1+\tan ^2\theta }{\csc ^2\theta }+\sin ^2\theta +\dfrac{1}{\sec ^\theta }\)
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\(\sec^2 x\)
14) \(\left (\dfrac{\tan x}{\csc ^2 x}+\dfrac{\tan x}{\sec ^2 x} \right )\left (\dfrac{1+\tan x}{1+\cot x} \right )-\dfrac{1}{\cos ^2 x}\)
15) \(\dfrac{1-\cos ^2 x}{\tan ^2 x}+2\sin ^2 x\)
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\(\sin^2 x+1\)
For the exercises 16-28, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
16) \(\dfrac{\tan x+\cot x}{\csc x}; \cos x\)
17) \(\dfrac{\sec x+\csc x}{1+\tan x}; \sin x\)
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\(\dfrac{1}{\sin x}\)
18) \(\dfrac{\cos x}{1+\sin x}+\tan x; \cos x\)
19) \(\dfrac{1}{\sin x\cos x}-\cot x; \cot x\)
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\(\dfrac{1}{\cot x}\)
20) \(\dfrac{1}{1-\cos x}-\dfrac{\cos x}{1+\cos x}; \csc x\)
21) \((\sec x+\csc x)(\sin x+\cos x)-2-\cot x; \tan x\)
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\(\tan x\)
22) \(\dfrac{1}{\csc x-\sin x}; \sec x\) and \(\tan x\)
23) \(\dfrac{1-\sin x}{1+\sin x}-\dfrac{1+\sin x}{1-\sin x}; \sec x\) and \(\tan x\)
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\(-4\sec x \tan x\)
24) \(\tan x; \sec x\)
25) \(\sec x; \cot x\)
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\(\pm \sqrt{\dfrac{1}{\cot ^2 x}+1}\)
26) \(\sec x; \sin x\)
27) \(\cot x; \sin x\)
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\(\dfrac{\pm \sqrt{1-\sin ^2 x}}{\sin x}\)
28) \(\cot x; \csc x\)
For the exercises 29-33, verify the identity.
29) \(\cos x-\cos^3x=\cos x \sin^2 x\)
- Answer
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Answers will vary. Sample proof:
\(\begin{align*} \cos x-\cos^3x &= \cos x (1-\cos^2 x)\\ &= \cos x\sin ^x \end{align*}\)
30) \(\cos x(\tan x-\sec(-x))=\sin x-1\)
31) \(\dfrac{1+\sin ^2x}{\cos ^2 x}=\dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x}=1+2\tan ^2x\)
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Answers will vary. Sample proof:
\(\begin{align*} \dfrac{1+\sin ^2x}{\cos ^2 x} &= \dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x}\\ &= \sec ^2x+\tan ^2x\\ &= \tan ^2x+1+\tan ^2x\\ &= 1+2\tan ^2x \end{align*}\)
32) \((\sin x+\cos x)^2=1+2 \sin x\cos x\)
33) \(\cos^2x-\tan^2x=2-\sin^2x-\sec^2x\)
- Answer
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Answers will vary. Sample proof:
\(\begin{align*} \cos^2x-\tan^2x &= 1-\sin^2x-\left (\sec^2x -1 \right )\\ &= 1-\sin^2x-\sec^2x +1\\ &= 2-\sin^2x-\sec^2x \end{align*}\)
Extensions
For the exercises 34-39, prove or disprove the identity.
34) \(\dfrac{1}{1+\cos x}-\dfrac{1}{1-\cos (-x)}=-2\cot x\csc x\)
35) \(\csc^2x(1+\sin^2x)=\cot^2x\)
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False
36) \(\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2x=\cos 2x\)
37) \(\dfrac{\tan x}{\sec x}\sin (-x)=\cos ^2x\)
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False
38) \(\dfrac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)\)
39) \(\dfrac{1+\sin x}{\cos x}=\dfrac{\cos x}{1+\sin (-x)}\)
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Proved with negative and Pythagorean identities
For the exercises 40-, determine whether the identity is true or false. If false, find an appropriate equivalent expression.
40) \(\dfrac{\cos ^2 \theta -\sin ^2 \theta }{1-\tan ^\theta }=\sin ^2 \theta\)
41) \(3\sin^2\theta + 4\cos^2\theta =3+\cos^2\theta\)
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True
\(\begin{align*} 3\sin^2\theta + 4\cos^2\theta &= 3\sin ^2\theta +3\cos ^2\theta +\cos^2\theta \\ &= 3\left ( \sin ^2\theta +\cos ^2\theta \right )+\cos^2\theta \\ &= 3+\cos^2\theta \end{align*}\)
42) \(\dfrac{\sec \theta +\tan \theta }{\cot \theta+\cos ^\theta }=\sec ^2 \theta\)