Secant, Cosecant, and CotangentDate: 01/16/97 at 18:20:41 From: Anonymous Subject: Trig Dear Dr. Math, I'm an eleventh grader and I have a problem understanding equations using the secant, cosecant, and cotangent functions. Is there any way you can help me? The problems I have difficulty with are like this: In the interval 0 <= x < 2pi, identify: A. all values at which the function is undefined B. all x-intercepts I thank you very much, Jim Hagler Date: 01/24/97 at 14:51:13 From: Doctor Toby Subject: Re: Trig Here is how I remember these things: First, remember the formulas for each function in terms of sine and cosine: tan = sin/cos cot = cos/sin sec = 1/cos csc = 1/sin If all you want to do now is solve problems A and B above, skip down to the row of asterisks below. The stuff before that may help you if you have other problems along similar lines, but it's not necessary for problems A and B. Memorize these facts about the sine and cosine (which perhaps you already have memorized): In the first quadrant (0 < x < pi/2), sine and cosine are both positive. In the second quadrant (pi/2 < x < pi), only sine is positive. In the third quadrant (pi < x < 3 pi/2), neither sine nor cosine is positive. In the fourth quadrant (3 pi/2 < x < 2 pi), only cosine is positive. Remember the order: both; sine; neither; cosine. Ultimately, these facts are true because the cosine is the x-coordinate and the sine is the y-coordinate of a point on a circle of radius 1 centred at the origin. If you ever need to know the sign of one of the other trig functions, just use the information you memorized about the sine and cosine. For example, what sign is the cotangent in the second quadrant? Cosine is negative there, while sine is positive there, so calculate: cot = cos/sin = -/+ = -; cotangent is negative there. ********************************************************************** Sine and cosine are both continuous functions defined everywhere. Therefore, whenever one of them changes sign, its value is zero there. So sine is zero at 0 and at pi, while cosine is zero at pi/2 and at 3 pi/2. The formulas for the other trig functions in terms of sine and cosine give these functions as fractions. A fraction is undefined whenever its denominator is zero; otherwise, it is zero whenever its numerator is zero. For example, cotangent = cosine/sine, so cotangent is undefined when sine is zero (at 0 and at pi) and zero when cosine is zero (at pi/2 and at 3 pi/2). You can calculate this for the other functions the same way. (Note that some functions might never be undefined or zero.) -Doctor Toby, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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