Solutions for The Other Trigonometric Functions
Solutions to Try Its
1.![A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.](http://cnx.org/resources/932dac4794332f5671db2989a9bec2e8bc3731d7/CNX_Precalc_Figure_06_02_004.jpg)
![A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.](http://cnx.org/resources/1b7052482b267dfd074c23a688b1021563d39ce7/CNX_Precalc_Figure_06_02_011.jpg)
![A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi/20 and one approximately at 3pi/16.](http://cnx.org/resources/ef79fb928275f270698992089f9ecd1e0b515b94/CNX_Precalc_Figure_06_02_013.jpg)
![A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.](http://cnx.org/resources/af909338dc0d6874ab180c82e22d89e2ece8cb66/CNX_Precalc_Figure_06_02_023b.jpg)
![A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.](http://cnx.org/resources/1e40f8b1eec1887261ecebfa1ca394723f32179d/CNX_Precalc_Figure_06_02_016.jpg)
Solutions to Odd-Numbered Exercises
1. Since [latex]y=\csc x[/latex] is the reciprocal function of [latex]y=\sin x[/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\sin x[/latex] to obtain the y-coordinates of [latex]y=\csc x[/latex]. The x-intercepts of the graph [latex]y=\sin x[/latex] are the vertical asymptotes for the graph of [latex]y=\csc x[/latex]. 3. Answers will vary. Using the unit circle, one can show that [latex]\tan(x+\pi)=\tan x[/latex]. 5. The period is the same: 2π. 7. IV 9. III 11. period: 8; horizontal shift: 1 unit to left 13. 1.5 15. 5 17. [latex]−\cot x\cos x−\sin x[/latex] 19. stretching factor: 2; period: [latex]\frac{\pi}{4}[/latex]; asymptotes: [latex]x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8[/latex], where k is an integer![A graph of two periods of a modified tangent function. There are two vertical asymptotes.](http://cnx.org/resources/1b56c51a5f20a9733b5f30e56da1004a0fa568c2/CNX_Precalc_Figure_06_02_202.jpg)
![A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.](http://cnx.org/resources/6e880e16184a36c8619e08545ed2283ff9338b22/CNX_Precalc_Figure_06_02_204.jpg)
![A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.](http://cnx.org/resources/2cb5d02f855749ed862df08c32e87c85c92d5162/CNX_Precalc_Figure_06_02_206.jpg)
![A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.](http://cnx.org/resources/f290423a0ace06ae1bbd748f0f25d214c4596101/CNX_Precalc_Figure_06_02_208.jpg)
![A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.](http://cnx.org/resources/d883b470c508727ae75cba2271e013941f825aae/CNX_Precalc_Figure_06_02_210.jpg)
![A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi/2, -pi/6, pi/6, and pi/2.](http://cnx.org/resources/184e940889552f60525e38a28e0b079eeb945ddc/CNX_Precalc_Figure_06_02_212.jpg)
![A graph of two periods of a modified secant function. There are four vertical asymptotes all pi/5 apart.](http://cnx.org/resources/983527f6197a4cad727909e1c5dc9c80b558f89a/CNX_Precalc_Figure_06_02_214.jpg)
![A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.](http://cnx.org/resources/6ba424d590a3269fb164a28a2adeba07b39e28fc/CNX_Precalc_Figure_06_02_216.jpg)
![A graph of a modified cosecant function. Four vertical asymptotes.](http://cnx.org/resources/0927a7fed94486f39191d9e2d95e13c99c6c8f8c/CNX_Precalc_Figure_06_02_218.jpg)
![A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12.](http://cnx.org/resources/83b78668cd8dfe9571ee25b737eaa491d773ee6e/CNX_Precalc_Figure_06_02_220.jpg)
![A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.](http://cnx.org/resources/0bb6095438602382c3cbebfceac5edf4d7cc98d5/CNX_Precalc_Figure_06_02_234.jpg)
![A graph of y=1.](http://cnx.org/resources/1174db8d0850b37bb1ca86bf78487bf885d1c876/CNX_Precalc_Figure_06_02_241.jpg)
![A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.](http://cnx.org/resources/d003975c56c76c0edb4ca5b1d697a0ff82ead133/CNX_Precalc_Figure_06_02_238.jpg)
![An exponentially increasing function with a vertical asymptote at x=60.](http://cnx.org/resources/1e380791f3f6e3024dfef7ae5212526a8d585f50/CNX_Precalc_Figure_06_02_240.jpg)
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.