Back to Study Notes
Study Notes 029:
Trigonometry 3:
By William E. Steinman:
Plane trigonometry is concerned with plane angles and triangles. It also deals with special functions of angles without reference to triangles.
First, let us consider ratios in similar triangles. Begin with a simple plane angle with rays of undetermined length.
We can place perpendiculars from one side to intersect the other side and form right triangles.
Now we can get to the six trigonometric functions. These functions are not derived, they are given. The trigonometric tables can then be derived using these functions. Once we have the tables, we can use them to computer various parameters of any given angle or triangle. We will get to that. First, the six functions.
The sin of an angle Ө = the side opposite/the hypotenuse.
In the above triangle that becomes sine Ө = ad/oa = be/ob = cf/oc.
The cosine Ө = side adjacent/hypotenuse = od/oa
The tangent Ө = side opposite/side adjacent = ad/od
The cosecant Ө = hypotenuse/side opposite = oa/ad
The secant Ө = hypotenuse/side adjacent = oa/od
The cotangent Ө = side adjacent/side opposite = od/ad
Normally, when we uses these functions we abbreviate them to sin Ө, cos Ө, tan Ө, csc Ө, sec Ө, cot Ө.
So let’s take the good old 3,4,5 right triangle to get the hang of these things. This a right triangle whose legs are 3 unit and 4 units respectively giving a hypotenuse of 5 units.
Work it out with Pythagoras’ theorem if you wish.
Remember?
The square of the hypotenuse of a right triangle is equal to the sum of the squares of its two sides.
We can use the above definitions to compute the various values.
Sin Ө = side opposite/hypotenuse = 3/5 = 0.6
Cos Ө = side adjacent/ hypotenuse = 4/5 = 0.8
Tan Ө = side opposite/side adjacent = 3/4 = 0.75
Csc Ө = hypotenuse/side opposite = 5/3 = 1.6666…
Sec Ө = hypotenuse/side adjacent = 5/4 = 1.25
Cot Ө = side adjacent/side opposite = 4/3 1.3333…
What can we do with these numbers?
We can use them to find the corresponding angles in the table of trigonometric functions.
Try it. Take the sin of 0.6
Look up the sin in the table, SN 027.
Whoops!
There is no value for the sin = 0.6.
What now?
It is really no problem to extrapolate between the values we can find in the table.
The value for 360 is 0.58779.
The value for 370 is 0.60182.
If we subtract the smaller value from the larger value we get a difference of 0.01403. Recall that we have 60 minutes in one degree of arc.
Thus we can divide our difference by 60 to get the value for each minute between these two values.
0.1403/60 = 0.00023 approximately.
Okay we can take the difference between 0.6 and our lower value.
That is 0.6 - 0.58779 = 0.01221.
We can divide that by the value for one minute to get the difference in minutes.
We will get approximately 53 minutes.
So our angle, to the nearest minute is 36053′ .
You could work it out to seconds if you so wished. I am satisfied with this value.
As we used to say when I was working for a government contractor, “It’s good enough for government work.”