Trig Problem: Sec A = -2 ...Date: 10/20/97 at 22:47:43 From: Jennifer Rausch Subject: math 30 I am having problems with the folowing problem: sec A = -2 where 0 < A < 2(pi) Where would I start? Help, please! Jennifer Date: 10/24/97 at 11:02:52 From: Doctor Bruce Subject: Re: math 30 Hi Jennifer, I'm going to guess that you want to know how to find all the angle(s) in the range between 0 and 2*pi radians that have a secant equal to -2. I've always found it a good thing to change trig problems so that everything is expressed in terms of sines and cosines. We know that secant(A) = 1/cosine(A). So, your problem is the same as finding all the angles A in the range between 0 and 2*pi radians that have a cosine equal to -1/2. When I took trigonometry (shortly after the last Ice Age ended), I was made to memorize the sines and cosines of a few "special" angles, namely 0,30,45,60, and 90 degrees. So, I can still reach into my memory and recall that cosine (60 degrees) = 1/2, or, the same thing in radians, cosine (pi/3 radians) = 1/2. You can certainly check that this is true with a pocket calculator. We got the number 1/2 all right, but we need an angle with cosine -1/2, not +1/2. Now comes the matter of looking at all the reflections of an angle in the four quadrants. For each trig function, two reflections give positive values and two give negative. I recall that angles in quadrants I and IV have positive cosines, and angles in quadrants II and III have negative cosines. So, the reflection of (pi/3) into both quadrants II and III gives two angles, each with cosine equal to -1/2. Since you gave the restriction that 0 < A < 2*pi, we don't have to go any further in solving this problem. (Maybe you have seen other problems where you have to add on extra multiples of 2*pi to find ALL the angles with a given cosine value. If the range for A were larger, you'd have to do that in this problem. But 0 < A < 2*pi means to look at just the angles in the four quadrants.) I hope that the counter-clockwise numbering I,II,III,IV of the quadrants is familiar to you. I doubt that any other numbering system is used. What I mean by reflecting an angle into another quadrant is this. Say the angle B is in the first quadrant. Then the reflection of B into the other quadrants is given by this little table. quadrant I II III IV reflected angle B pi - B pi + B 2*pi - B Now, I didn't finish working out the problem for you - we Math Doctors are supposed to leave you a little work to do on your own! But if you don't manage to come up with the answer, write back and we'll try to give some more hints. -Doctor Bruce, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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