TRIGONOMETRY

TRIGONOMETRY

Cosine, Sine and Tangent are three of a series of mathematical functions used in the study of right angled triangles (any triangle where one angle is at 90 degrees (see figure 1). This is part of the branch of maths known as Trigonometry - the study of all triangles.

These three functions represent the ratio between two of the sides of a right angled triangle for a particular angle. For any particular angle, this ratio will always be the same, regardless of the actual size of the triangle. If you know the angle of one other corner of the triangle, and the length of any one side of the triangle, you can calculate the length of the other two sides. There are several different ways of measuring angles, of which degrees are the most familiar, and will be used here. Poser uses the Radian, which is much more elegant mathematically. The basic principles of trigonometry are the same regardless of which method you use to measure angles.


Fig 1: The Parts of the Triangle

The Parts of the Triangle

To understand what's going on in these functions, we need to have clear names for each part of the triangle. Figure 1 and the list below shows the four most important terms used here.

b = the angle

h = hypotenuse: the long side, opposite the right angle

o = opposite: the side opposite to the angle 'b'

a = adjacent: the short side next to the angle 'b'

Sine, Cosine and Tangent

For any given angle of b, we can calculate the sine, cosine and tangent by measuring the sides of the triangle and performing the following calculations. For our example, we will use a triangle where a = 1, o = 2, h = 2.236 and b = 63.434 degrees.

sin (b) = o / h

sin (63.434) = 2 / 2.236 = 0.894

cos (b) = a / h

cos (63.434) = 1 / 2.236 = 0.447

tan (b) = o / a

tan (63.434) = 2 / 1 = 2

It doesn't matter how big the triangle is - as long as the angle b stays the same then the ratio between the sides of the triangle will remain the same, and the results of the sine, cosine and tangent calculations will remain the same (for example if a=1.5, then o=3, resulting in tan(63.434)= 3 / 1.5 = 2).

Although there are mathematical ways to calculate the values of sine, cosine and tangent just from the angle, they are complex and time consuming and we don't need to use them. Poser provides us with maths functions that do the work for us (see below for how the trig. functions were used in the past).

Derived Functions

Now we have these three functions, we need to modify them so that we can use them when we know 'b' and one other value. We can do that using some simple mathematical tricks. When we have a formula like sin (b) = o / h we can perform maths on the formula as long as we do the same thing to both sides of the formula.

First of all, we will multiply both sides by h. That will turn sin (b) into sin (b) * h. On the other side, o / h is turned into just o (if you take any number, divide it by another number and then multiply it by the same number you will always return to the original number). This gives us a new formula: sin (b) * h = o. Finally, swap the sides around, and we get o = sin (b) * h. If we know the angle and the hypotenuse we can use this new formula to work out the opposite.

We can go one step further. o = sin (b) * h can also be written as o = h * sin (b). If we divide both sides of this formula by sin (b) we end up with o / sin (b) = h. Swap that around and we get h = o / sin (b).

Table 1 shows all six formula that can be created in this way.

TABLE 1: Sine, Cosine and Tangent formula.

sin (b) = o / h o = sin (b) * h h = o / sin (b)
cos (b) = a / h a = cos (b) * h h = a / cos (b)
tan (b) = o / a o = tan (b) * a a = o / tan (b)

Table 2 shows which formula to use in each possible set of circumstances.

TABLE 2: Which formula to use when

  Have a Have o Have h
Need a - o / tan (b) cos (b) * h
Need o tan (b) * a - sin (b) * h
Need h h = a / cos (b) o / sin (b) -

Using Sine, Cosine or Tangent

The standard example used to explain when you might want to use trigonometry is the measurement of heights. Let us say that you need to know the height of the top of your house. Actually measuring up the side of the building would be awkward to say the least. Instead, we will use our trigonometry.

First, pick a spot a little way away from the building. Measure the distance from that spot to the base of the house wall. This will be our 'a' or adjacent. In this example, we ended up 10 meters away from the house.

Second, from your spot measure the angle between the ground and the highest point on the building. This becomes our 'b' or angle. In this example, the angle from our spot 10 meters away from the house to tip of the roof was 30 degrees.

We now have a and b, and we need to calculate 'o', the opposite. If you look at the list of functions we have above, you will see that o = tan (b) * a.

That means that 0 = tan (30) * 10.

Tan (30) is equal to 0.577. Therefore our house is 0.577 * 10, or 5.77 meters high.

Calculating Sine, Cosine or Tangent.

Prior to the advent of the scientific calculator and the computer, if you needed to find out a value for sine, cosine or tangent, you would use a lookup table. This was simply a pre-calculated list of the values of the trigonometry functions for a particular set of angles. If you needed an angle that isn't on the list, you could estimate it from the list. On our sample table below, we have values for 1 and 2 degrees. If we wanted sine(1.5), first we would look at the values for 1 and 2 and calculate the gap between them (0.0349-0.0175= 0.0174). Next, we would look at the fraction of our angle (.5), and multiply our 0.0175 by 0.5, to get 0.0087. Finally, we would add that to the value for 1 degree, reaching an approximation of 0.0175+0.0087= 0.0262. This is actually accurate to the four decimal places we see - the windows Calculator returns sine(1.5)= 0.026176948307873152610611685554113 - if we round this up to 4 decimal places, then the final result is 0.262. If you were working in a field that needed more accuracy that this (aircraft design for instance), then you'd buy a more detailed set of lookup table.

Radians Degrees Sine Cosine Tangent
.0000 0 .0000 1.000 .0000
.0175 1 .0175 .9998 .0175
.0349 2 .0349 .9994 .0349

Even many modern computers use lookup tables, combined with more accurate formulas for filling the gaps, to calculate sine, cosine and tangent.


PoserWorld