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Study Notes 036:
Trigonometry 10:
By William E. Steinman:
Let’s look at what we call the inverse trigonometric functions.
Let us begin with something that seems simple.
x = sin y
We see that for every value of y given, there will be a unique value for x.
Simple enough.
However, the inverse is not so simple.
For a given value of x there may be no solutions as in x = 35.
The sin can never exceed a value of 1.
There may also be many solutions for a given value of x.
For example, if x = 0.5, y can be 300, 1500, 3900, etcetera.
So far, so good!
Okay.
Now let’s look at y = arc sin x.
This is not what it looks like.
By convention, the statement y is the sine of x, or y = sin x is equivalent to the statement x is an angle, the sine of which is equal to y.
This is written symbolically as x = arc sin y = sin-1 y.
In this case, -1 is not an exponent, but part of the inverse symbol.
If we wanted to indicate an exponent we would write is as (sin y)-1.
To avoid confusion, the arc form is preferred.
The inverse functions, arc cos y, arc tan y, arc cot y, arc sec y, arc csc y, are also defined in this same way.
We are still faced with the fact that we have many solutions to the equation
y = sin x.
To deal with that, we can specify the principle values of the inverse functions.
In our example above it is useful to notice that only one of the possible values is a positive acute angle, that being 300.
Therefore we can designate the positive acute angle of 300 as the principle angle whose sine is 0.5.
Now we will need a way to specify any angle other than the principle angle.
Our convention for that is to capitalize arc.
For example y = Arc sin x means any angle whose sine is x.
One thing I used to hate in formal education is the fact that most of the stuff I was being taught was not related back to real situations. It was taught in a very abstract way.
I am not teaching, but sharing what I am learning, so I can feel safe in sharing how trigonometry relates to the world we live in.
At one time, I also wondered what some of my readers are wondering.
Namely, “Of what worldly use can this stuff be?”
I will give you a generalized answer from my own experience.
The trigonometric functions are very useful in many braches of engineering because many actions in nature tend to behave as wave phenomena that, in ideal situations, exactly duplicates the trigonometric functions.
One obvious example is the wave behavior of a liquid surface when it is disturbed, such as tossing a rock in a pond.
Anther example, though you cannot see it, is the wave action of alternating electrical current.
Let’s consider that.
Look at the equations y = sin x
If we plot this equation we will plot the sine of the angle with respect to the angle and produce a drawing of what is called a sin wave.
Here is what it looks like. This is a drawing I copied from Microsoft’s Encarta.
If we picture a radius vector going around a circle, this plot would represent one complete revolution of that vector. The vector would move from 00 through 900, through 1800, through 2700, to 3600, which is also 00.
Now consider a simple electrical generator, which would produce AC current.
If we plot the current amplitude as a function of the rotation of the generator it would look exactly like the curve above. Alternating current obeys the trigonometric functions. This is how an electrical engineer would see it.
That is simply an example. This is a study of trigonometry, not electricity, so we can leave it there.
That is enough for this time.
I suggest you copy this page and go over it again tomorrow.