Here is a trig table based on a circle of radius 200. Angles are given in degrees. You can tell that the circle is of radius 200 because the table gives sin 90 as 200.
|
Angle |
Sin |
Tan |
Sec |
|
0 |
0 |
0 |
200 |
|
10 |
35 |
35 |
203 |
|
20 |
68 |
73 |
213 |
|
30 |
100 |
115 |
231 |
|
40 |
129 |
168 |
261 |
|
45 |
141 |
200 |
283 |
|
50 |
153 |
238 |
311 |
|
60 |
173 |
346 |
400 |
|
70 |
188 |
549 |
585 |
|
80 |
197 |
1134 |
1152 |
|
90 |
200 |
|
|
1. How does sin 90 = 200 tell you that the circle is of radius 200?
2. How can you use this table to get our modern value of sin 45, which we know to be about 0.701?
3. How can you use this table to get cosines?
4. In this table, is it still true that tan x = (sin x)/(cos x)? How about sec x = 1/cos x? Why or why not?
5. Suppose ABC is a right triangle with right angle at C, and that sides a, b, c are opposite angles A, B, C, respectively. If b = 25 and if A = 40°, then use the table to find the length of side a. You should not convert to modern values of sine and cosine. In fact, it is better if you do not use any decimals, only fractions and whole numbers.
6. Suppose that ABC is a right triangle, with A = 20° and c = 40. Find a.
1. If a segment of length r is at an angle A to a base, and the top of the segment is at distance r sin A from the base, using modern sines. We’re given that A = 90° and that the distance is 200. This makes r = 200 as well.
2. We can divide 141, the given value for sin 45, by the "total sine", 200, and get 0.705, which is correct to two decimal places. In fact, all the values in the table are 200 times their usual values.
3. It is still true that cos A = sin (90°- A). So, for example, cos 40° = sin (90°- 40°) = sin 50° = 153.
4. For this calculation, let’s use capital letters for the functions given in the table, and lower case letters for the modern functions. Then SIN x/COS x = (200 sin x)/(200 cos x) = tan x = (TAN x)/200, and this is not equal to TAN x, so the statement is not true. A student at the time would have learned to write TOT SIN for the radius of the circle involved, and then learned trigonometric identities like SIN x/COS x = TAN x/TOT SIN. That student would have learned it in ratio form, though, so the identity would have looked like SIN x : COS x :: TAN x : TOT SIN.
Similarly, 1/COS x = 1/(200 cos x) = (sec x)/200 = (SEC x)/(2002) = (SEC x)/((TOT SIN)2), and this is not equal to SEC x. Again, the proposed statement is not true.
5. You might start with a/b = TAN A/TOT SIN. This makes 168/200 = a/25, so a = 25•168/200 = 21. Modern methods give a value of 20.9775.
6. You might start with a/c = SIN A/TOT SIN. This gives a/40 = 68/200, so a = 68•40/200 = 13 3/5. Modern methods give 13.6808