Looking at the values of Trigonometric functions

by Amy Benson

Every time hit a trig button on my calculator, the calculator "knows" the answer and responds appropriately. As I am constantly encouraging my students not to rely on their calculators for everything (after all a machine is only as good as its operator), I want to find the values of the sine, cosine, and tangent of 5 degrees through 90 degrees in increments of 5 degrees.

I begin with the basic right isosceles triangle

Recall that

and and

So,

If I am to compare these values to the values I obtain from my calculator, I must approximate:

Now, I shift my focus to an equilateral triangle with sides of length 1 unit. I will use this triangle to find the values of the trigonometric ratios for 30 degrees and 60 degrees.

Using the Pythagorean Theorem, I find that the perpendicular bisector (height of the triangle) of one side of the triangle is .

So,

while

Again, approximation is required:

and and and, of course.

At this point, in my computations, I need a few helpful formulas:

With these formulas, I will find compute the sine, cosine and tangent of , , and .

Starting with , I use the half-angle formula to find the cosine.

Having found the cosine, I easily find the sine.

Now, the tangent:

In order to find the values of the trigonometric functions for , I use the values I found when x=and

x=.

and

Next, I look at x=.

and

and

To go any further, I will ahve to use the sine power series. I begin this portion of my computations I use

To compute sin , I use a TI-82 graphing calculator. First, I convert to and store this value as x. Then, I program the calculator to sum the series accurate to nine decimal places. The steps are as follows:

yields

.0871557006

.0871557428

hit enter 16 times

.0871557427

So, I know continue my computations confident that the .

I use my newly found value to find the value of cosine:

and the tangent value:

All subsequent computations to find the values of the three trignometric functions in question involve the use of aforementioned trignometric formulas. Hence, if you wish to see those computations, click here. Otherwise, here is the table of values:

 x sin(x) cos(x) tan(x) .08716 .99620 .08749 .17368 .98481 .17633 .25882 .96593 .26795 .34202 .93969 .36397 .42262 .90631 .46631 .50000 .86603 .57735 .57358 .81915 .70021 .64279 .76604 .83910 .70711 .70711 1.0000 .76604 .64279 1.19175 .81915 .57358 1.42815 .86603 .50000 1.73205 .90631 .42262 2.14451 .93969 .34202 2.74748 .96593 .25882 3.73205 .98481 .17368 5.67128 .99620 .08716 11.43005 1.0000 0.0000 undefined

In order to complete a similar table with angles by increments of instead of , use the sine/cosine power series to find the trignometric values at and then systematically use the formulas used in this essay to find all subsequent values.