In the right triangle OAB we shall let AB = y, OA = x, OB = r, thus adopting the letters of higher mathematics. Then, so long as ∠O remains the same, such ratios as y/x, y/r, etc., will remain the same, whatever is the size of the triangle. Some of these ratios have special names. For example, we call
y/r the sine of O, and we write sin O = y/r;
x/r the cosine of O, and we write cos O = x/r;
y/x the tangent of O, and we write tan O = y/x.
Now because
sin O = y/r, therefore r sin O = y;
and because cos O = x/r, therefore r cos O = x;
and because tan O = y/x, therefore x tan O = y.
Hence, if we knew the values of sin O, cos O, and tan O for the various angles, we could find x, y, or r if we knew any one of them.
Now the values of the sine, cosine, and tangent (functions of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:
| Angle | Sine | Cosine | Tangent | Angle | Sine | Cosine | Tangent |
| 5° | .087 | .996 | .087 | 50° | .766 | .643 | 1.192 |
| 10° | .174 | .985 | .176 | 55° | .819 | .574 | 1.428 |
| 15° | .259 | .966 | .268 | 60° | .866 | .500 | 1.732 |
| 20° | .342 | .940 | .364 | 65° | .906 | .423 | 2.145 |
| 25° | .423 | .906 | .466 | 70° | .940 | .342 | 2.748 |
| 30° | .500 | .866 | .577 | 75° | .966 | .259 | 3.732 |
| 35° | .574 | .819 | .700 | 80° | .985 | .174 | 5.671 |
| 40° | .643 | .766 | .839 | 85° | .996 | .087 | 11.430 |
| 45° | .707 | .707 | 1.000 | 90° | 1.000 | .000 | ∞ |
It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5° is 0.99619, and the sine of 85° is 0.99619. The entire table can be copied by a class in five minutes if a teacher wishes to introduce this phase of the work, and the author has frequently assigned the computing of a simpler table as a class exercise.
Referring to the figure, if we know that r = 30 and ∠O = 40°, then since y = r sin O, we have y = 30 × 0.643 = 19.29. If we know that x = 60 and ∠O = 35°, then since y = x tan O, we have y = 60 × 0.7 = 42. We may also find r, for cos O = x/r, whence r = x/(cos O) = 60/0.819 = 73.26.
Therefore, if we could easily measure ∠O and could measure the distance x, we could find the height of a building y. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. We may then measure the distance to the building and apply the formula y = x tan O.