============================================================ ANGLE | SINE |COSINE|TANGENT|| ANGLE | SINE |COSINE|TANGENT ------+------+------+-------++-------+------+------+-------- 5 deg. | .087 | .996 | .087 || 50 deg. | .766 | .643 | 1.192 ------+------+------+-------++-------+------+------+-------- 10 deg. | .174 | .985 | .176 || 55 deg. | .819 | .574 | 1.428 ------+------+------+-------++-------+------+------+-------- 15 deg. | .259 | .966 | .268 || 60 deg. | .866 | .500 | 1.732 ------+------+------+-------++-------+------+------+-------- 20 deg. | .342 | .940 | .364 || 65 deg. | .906 | .423 | 2.145 ------+------+------+-------++-------+------+------+-------- 25 deg. | .423 | .906 | .466 || 70 deg. | .940 | .342 | 2.748 ------+------+------+-------++-------+------+------+-------- 30 deg. | .500 | .866 | .577 || 75 deg. | .966 | .259 | 3.732 ------+------+------+-------++-------+------+------+-------- 35 deg. | .574 | .819 | .700 || 80 deg. | .985 | .174 | 5.671 ------+------+------+-------++-------+------+------+-------- 40 deg. | .643 | .766 | .839 || 85 deg. | .996 | .087 |11.430 ------+------+------+-------++-------+------+------+-------- 45 deg. | .707 | .707 | 1.000 || 90 deg. | 1.00 | .000 |[infinity] ============================================================ It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5 deg. is 0.99619, and the sine of 85 deg. 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 [L]_O_ = 40 deg., then since _y_ = _r_ sin _O_, we have _y_ = 30 x 0.643 = 19.29. If we know that _x_ = 60 and [L]_O_ = 35 deg., then since _y_ = _x_ tan _O_, we have _y_ = 60 x 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 [L]_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_. [Illustration: A QUADRANT OF THE SIXTEENTH CENTURY Finaeus's "De re et praxi geometrica," Paris, 1556]