Trigonometry in a Nutshell

The problem addressed by trigonometry is that of describing the relations betweens angles and side lengths in a triangle.

If two line segments, L and M, end at a point P they determine two angles at that point. One is the angle clockwise from L to M and the other the angle counterclockwise from L to M.

The first question to consider is: how do we measure such angles? That is, how do we assign a number to either of these angles?

There are several different ways to measure angle, and trigonometry is concerned with the relations between them.

The first way is the additive way: that is, we define angles at P so that if you put one angle right next to the other, so that one is between L and M and the other between M and N, then the angle between L and N is the sum of the angles between L and M and between M and N.

(this is a good place for a picture)

If we define angles this additive way, the only thing left to do is to define the unit of angle. All other angles follow by additivity from this unit value.

Naturally we have two different standard ways to define the unit of angle: historically the angle that goes all the way around, from L to L the long way, was called 360 degrees.

Why?

I have no idea.

Another way is to measure the angle between L and M at P by the distance along a unit circle centered at P between the two line segments. This distance is the standard measure of angle in units called radians. Since the circumference of a unit circle is 2, we can identify an angle of 360 degrees as one of 2 radians.

Thus a "straight line" angle, for which L and M are in opposite directions, is half of this or 180 degrees, or radians. A right angle is half of a straight line angle, and is therefore 90 degrees or radians. The small angle between the x-axis and the "main diagonal" (x = y) is half of this or 45 degrees or radians.

These definitions are all well and good, but distance around a circle is a bit awkward to measure, and there is an alternative way to describe angles. It is not additive, but much easier to measure in practice. It is called the sine of the angle, denoted as sin, and is 0 for the 0 angle and 1 for a right angle.

Suppose we are interested in the angle counterclockwise from M to L at point P. Imagine that we draw a unit circle around P and that L and M meet this circle at points A and B respectively. The sine of this angle is then defined to be the distance from the point A to the line M. The sine is the fundamental entity of trigonometry. You can measure it by drawing a line from A to M perpendicular to M and measuring its length.

There is one tiny complication: if the angle is bigger than a straight line angle, so that the angle counterclockwise from M to L is smaller than that clockwise, we define the sine of it to be negative.

(A picture here would help.)

The people who worked on such things long ago made lots of other definitions, which made perfect sense to them, but cluttered up the subject with lots of definitions, the memorization of which has hindered students of it ever since.

Why did they do that?

These definitions refer to obvious geometric quantities that they considered worth defining.

How so?

Well, they drew a line segment from the point A tangent to the unit circle around P to the line M. The length of this segment they called the tangent of the angle from M to L. (When the line has a positive slope the tangent is taken to be negative.)

The length of the line segment from P along M to the intersection of M with that tangent line at A they called the secant of the angle from M to L. (The secant is negative when this line runs away from M at P.)

They also defined the complement of an angle that is less than a right angle to be the difference between a right angle and it. This got them to define the cosine, cotangent and cosecant as the sine, tangent and secant of the complement of the original angle.

Fortunately for us, all of these six functions are easily related to the sine function, which means that we need only really become familiar with the sine, and we can then figure out what the others will do. Notice that the sine, by these definitions changes sign when we interchange L and M, while cosine stays the same.

Here are the relations between these functions, all of which follow from the definitions from the fact that corresponding angles of similar triangles are equal

Exercises:

1. Draw a relevant picture for an angle that is less than showing all of these entities. Include the line perpendicular to M at P, and its intersection with the tangent to the unit circle at A.

2. Identify which triangles are similar to one another. Remember that the two angles other than a right angle in a triangle with a right angle are complementary.

3. Deduce all the claims above by using the similar triangles argument mentioned above.

There are three basic theorems of trigonometry that you should know. Then there are the "addition theorems" of trigonometry and the relation between the sine and the exponential function, and you know all you should know about the subject.

OK what are the basic theorems?

1.The Pythagorean Theorem: This famous result states that the square of the hypotenuse of a right triangle is the sum of the squares of its other two sides. Translated to our definitions it says that for any angle, we have

which implies that, up to sign we have

2. The Law of Sines: This states that in any triangle ABC the ratio of the sines of its angle at A to its angle at B is the ratio of the lengths of the side opposite A to the side opposite B. If we describe these lengths as l(BC) and l(AC) respectively, we have

This statement follows if we drop a perpendicular from C to the line AB and relate the length of that perpendicular in terms of the angle B and the length of BC and also in terms of the angle A and the length of AC.

3. The Law of Cosines: This statement gives the length of the side BC of a triangle in terms of the lengths of AB and AC and its angle at A

l(BC)2 = l(AB)2 + l(AC)2 – 2 l(AB)*l(AC)*cos A

This result can be deduced in several ways. One way is by dropping a perpendicular from B to AC meeting the latter line at Q. We then have

l(AQ) = l(AB)*cos A, AC = AQ + QC, l(AQ)2 + l(BQ)2 = l(AB)2

and

l(CQ)2 + l(BQ)2 = l(BC)2

Appropriate substitution among these equations yields the stated law.

What in the world is an addition theorem?

We have already noticed that the standard measure of angle, in terms of degrees or radians is additive: this measure of the sum of two angles is the sum of the same measures of each summand. This statement is not true for sines or cosines. The sine of the sum of two angles is not the sum of their sines. The addition theorems tell us how to compute the sine and cosine of the sum of two angles in terms of the sines and cosines of the two angles that are summed.

And what are they?

Addition Theorems: Suppose we have two adjacent angles, of sizes q and j. Then the sine and cosine of their sum, q+ j are given by

And where do these claims come from?

Perhaps the easiest proof comes from the next discussion: the relation between sines and cosines and the exponential function. They follow quickly from that relation, on substitution, given the fundamental properties of the exponential function.

Exercises:

4. Write the corresponding results for the sines and cosines of and also for sines and cosines of 2.

5. Combine the last of these with the Pythagorean theorem to get expressions for and for in terms of cos.

 

Sines Cosines and Exponents

Given any function, f, we can define two others that are symmetric and antisymmetric under reflection about the origin, as follows

In words, g(x) is the average f(x) and f(-x), while h(x) is the average of f(x) and –f(-x). We can deduce from these definitions that g is symmetric and h antisymmetric (which means it changes sign under this reflection). We may also notice, f = g + h.

In light of these facts, we can call g the symmetric part of f and h the antisymmetric part of f, under reflection about the origin.

The symmetric part of the exponential function, ex is called cosh x, while the antisymmetric part is called sinh x. These are called the hyperbolic cosine and sine respectively, and you may have noticed corresponding buttons on your calculator.

If we consider the exponential function of an imaginary argument, eix we find that the symmetric part is real, while the antisymmetric part is imaginary. In fact the symmetric part of eix is cos x, and the antisymmetric part of expix is i sin x.

These facts, and the fundamental property of exponents: eAeB = eA+B, which is an addition theorem for exponents, provide for straightforward deduction of the addition theorem for sines and cosines.

The formal statements of the relation between sines and cosines and exponentials are as follows

Exercise 6 The exponential function being its own derivative, can be written as an infinite series as follows

Write out the first three terms of the series for sin x and for cos x implied by the relations just above and this statement.

Here is an applet to help you get used to these concepts.