More Algebra Hints

8. Trigonometry and Complex Numbers 

A. Trigonometry

The simplest way to introduce trigonometric functions (functions on your calculator) is to begin with their unit circle definitions, and then specialize to their right triangle computation with the help of  similarity assumptions about  triangles, right or scalene. Several steps follow for reading in or besides your trig course.

Step 1.

 Draw a unit circle

Your unit of measurement may be one centimeter, one meter, one kilometer, one inch, one foot, one yard, one mile or any other unit. Choose one,  or draw a circle and declare its radius to be your unit length.

Exercise for Later: How does similarity assumptions for right triangles imply the results, here the definition of trig functions below, is independent of the choose of unit length?

Step 2.

Let q be an angle. Locate the head of the vector with angle q and length 1 on the unit circle.

Step 3.

The head will have coordinates (a units, b units)

  on circle of radius 1 unit.

Put cos(q) =a and sin (q) =b. This defines both sine and cosine for all values of the angle q.

Further trig functions may be defined as follows.

when the divisors are nonzero.

The case where q is between 0 and 90 degrees is considered next.

Step 4 (Right Triangle Trigonometry)

circle of radius 1 unit.

Assume q is between 0 and 90 degrees.  Then

For angles between 0 and 90 degrees, similarity of right triangles implies the ratios

if you  replace the unit circle right triangle by a similar right triangle.

The latter  formulas for  may be used to compute with any right triangle where sides are labeled opposite and adjacent for an angle    The  further trig functions may be defined as follows.

when the divisors are nonzero.

Exercise: Express these further trig functions as ratios of the sides opposite, adjacent and/or hypotenuse of the above right triangle.

A trig course will explain the following  in more detail.

Trig functions link the ratio of two sides of a right triangle to cosines, sines and tangents of an angle. Knowledge of two sides in right triangle gives knowledge of the third by means of Pythagorean theorem, and of the values of the trig functions for the angles in the triangle.  Computation of unknown side lengths, unknown hypotenuse lengths and unknown angles is useful in land measurement (geo - metry) and also in navigation.

From one-to-one properties of trig functions for angles between 0 and 90 degrees or ½p, one can define (say how to compute) inverse trig functions (more functions on your calculator)  to compute the angles from the ratio of sides. Computation with inverse trig functions allows one to obtain polar coordinates for vectors or complex numbers from coordinates, real and imaginary parts, or the length of the adjacent and opposite sides of a right triangle determined by the coordinates. Again, this removes the need to measure the lengths and angles for points with rectangular coordinates [a, b].

Calculation

One may define trig functions by saying how to compute them in principle as above, but then one computes or approximates them in practice from tables and slide rules (old fashioned approach) or using calculators (the new approach). Unfortunately in this practice,  the tables, slide rules or calculation devices are black boxes which provide results, but whose derivation or justification is not commonly known. This departs from the principle of understanding the computations one does, but the numbers computed by these black boxes can be checked in simple cases. When calculators first arrived, some used faulty or suboptimal methods (algorithms) to compute.

B. Calculus

Geometric and Algebraic previews of calculus may help senior high school studies in and before calculus.

Calculus in the first instance is the subject of slope related computations, their reversal and interpretations. Calculus is the first course in which the algebraic way of writing and reasoning is required at full strength in several different ways.

C. Trigonometry and Complex Numbers

If z = (r,q) in polar coordinates, then   z  = a + i b =[a,b] = [r cos(q), r sin(q)] in rectangular coordinates. So the ability to compute cosines and sines avoids the need to measure the rectangular coordinates after a diagram after locating the point z from its polar coordinates.

A quick way, or the quickest way,  to understand and fully explain the algebraic properties of  trig functions is online is to start a trig courses after this treatment of complex complex numbers 

Logs, Powers and Exponentials of Complex Numbers
Preview of Electrical Engineering Mathematics

This last section defines (states formulas for) the exponential, logarithms and powers of complex numbers x+iy etc. If you are a science and engineering student you will eventually meet these functions and see their properties. This chapter gives a list of functions which you should expect to meet and understand in the first two years of your university studies. (The further discussion of these functions is left to a second or third course on calculus. From time to time, you should refer to the definitions given below to see how many on this list remain to be seen in your courses.)

The exponential of a complex number x+iy is given by

exp(x+iy) = ex[cos(y)+isin(y)] = ex cis(y)

What do you get if y = 0. What do you get if x = 0? 

Further, if x+iy = r cos(q) +isin(q) ¹ 0 with - = -180^° < q £ 180^° = p then the principal value of the natural logarithm

Note from the above definitions and algebraic properties of ln(x) and exp(x) for real numbers x,

• fundamental properties of exponentials: exp(z₁+z₂) = exp(z₁)exp(z₂)
• fundamental property of logarithms: ln(z₁z₂) = ln(z₁)+ln(z₂)+i2pn for some integer n Î {0,±1},
• first inverse property: exp(ln(z)) = z if z ¹ 0,
• second inverse property: ln(exp(z))-z = 2npi = ni 360 degrees for some integer n

Quick Definitions

• powers defined: z^x+iy = exp((x+iy)ln(z)) for z ¹ 0,
• the definition of a logarithm to the complex base a+ib:

  loga+ib(z) = ln(z) ln(a+ib)

• the hyperbolic cosine of the complex number x+iy defined:

  cosh(x+iy) = exp(x+iy)+exp(-x-iy) 2 What do you get if y = 0? What do you get if x = 0? Calculators give cosh(x)

   
• The hyperbolic sine of the complex number x+iy defined:

  sinh(x+iy) = exp(x+iy)-exp(-x-iy) 2 What do you get if y = 0? What do you get if x = 0? Calculators give sinh(x)

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