[H2_Conics] The mathematician's pool table

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[H2_Expository] Derivative of the Arcsecant function

Introduction

     This article explains how to differentiate, i.e. find the derivative of, the arcsecant function, which is seldom discussed in class.  If you look at the various syllabus outlines, sometimes they do not explicitly mention or imply the arcsecant, neither do they explicitly exclude this.  The important thing is: we should be able to figure it out from our basic knowledge, which is the whole point of mathematics.  The people who talk about and focus on mathematics syllabus content as if it is the only or most important thing are missing the point of mathematics education, and there are a plenty of these idiots around.  Do not follow them.

     The arcsecant, written as arcsec or sec^-1, is also known as the inverse secant function.  So arcsecant means that, given the value of a secant function, you want “the” angle whose secant is that given value.  The problem is there are many possible values.  Look at the graph below.

Defining the Inverse of the secant properly

     On the graph, a horizontal line can pass through (infinitely) many points.  Like any periodic trigonometric function, the secant function is not a one-to-one (a.k.a. “injective” or  “one-one”) function.  As such, it is not invertible.  However, we can restrict the function so that its domain is  [0, p] \ { ^p/₂ }.  This is highlighted in yellow on the graph.

     Remember that sec x = ¹/_{cos x}.  Basically we follow the principal values  [0, p]  of the arccosine function   except  ^p/₂  where the cosine is zero and its reciprocal the secant is undefined.       With this restriction on the domain, we get a one-one secant function, with range (-¥, -1] È [1, +¥).  We can now define the inverse function, and its graph is obtained by reflecting the above graph along the mirror line  y = x.  We get this:-

     Observe that in the yellow regions in both graphs, the gradients at the points are non-negative.  We have chosen the domain of the secant function, which is the same as the range of the arcsecant function, such that the derivatives will be non-negative.

Deriving the Derivative (refer to the“Onion” Method for differentiation)

Remarks

     Do not confuse  arccos y   with  (cos y)^-1.  They are not mean the same thing.

     In case you are wondering, the prefix “arc-” means the angle, which (if you use radians) is literally the same as the arc-length when the radius equals to  1.  In symbols, s = rq  with r = 1 means  s = q.  Although the term can be used with degrees or other units, when doing advanced mathematics like calculus, we would usually be using radians anyway. 

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics

* International Baccalaureate (IB) HL Mathematics

* Advanced Placement (AP) Calculus AB & BC

* University / College calculus

* other syllabuses that involve differentiation

* any learner interested in calculus

[JCH2CNEFTG_20150410] Exponential Half-Power Trick

Question

Introduction

     This is a complex-number question that appeared for ‘A’ Level Mathematics in November 1998.  Remember that mathematics is never out-dated.  Many Singapore schools keep this sort of questions in their question banks (for tutorial exercises, tests and examinations), in the hope that it becomes part of students’ repertoire.

     Students are expected to know the famous Euler’s Formula, one of the most beautiful formulas discovered by this visually-challenged but prolific mathematician.  It links the exponential function with trigonometry via the idea of angle rotation.  Adding e^if  with its reciprocal  e^-if (which is also its conjugate) gives a cosine expression, while subtracting gives a sine expression.

     The above question can be solved by rationalising the denominator and using heavy trigonometry and half-angle formulas.  There is nothing wrong with this approach.  I am going to illustrate a kewl approach, using what I call the exponential “half-power trick”.  Basically, whenever you see an expression like  1 ± e^{2kq i},  force out the factor  e^{kq i}.  This gives you either a sin or cos expression.  For example, e^{6q i} – 1 = e^{3q i} (e^{3q i} – e^{-3q i}) = i×2e^{3q i} sin q.

Solution

Observe that we have killed two birds with one stone.  At the last step, we just compare the real and imaginary parts to read off the answers.

Suitable Levels

* GCE ‘A’ Level H2 Mathematics (“Complex Numbers”)

* precocious students who love complex numbers