Consider the angle DA E, Fig. 152. From any point on the line AD drop a line perpendicular to the side AE forming the right triangle ABC. Let a represent the value or length of the side B C; let b represent the value of the side AC; let c represent the value of the side AB. The ratio of the side a to the side c is called the sine of the angle A. More concisely stated, a/c = sin A. The sine of an angle is the ratio of its opposite side to its hypotenuse, or opposite side over hypotenuse = sine of angle A = sin A. In a similar manner: Fig. 152.

 b = adjacent side cosine of angle A = cos A. c hypotenuse a - opposite side - tangent of angle A = tan A. b adjacent side b - adjacent side - cotangent of angle A = cot A. a opposite side c - hypotenuse = secant of angle A = sec A. b adjacent side c = hypotenuse cosecant of angle A = csc A. a opposite side

These ratios are known as natural functions of the angle because their values change with every change in the value of the angle.

The lengthening of the sides of the angle should not be mistaken for a change in the value of the angle. Draw to scale very carefully any angle and drop lines from any two points, as at B and B', Fig. 152, which shall be perpendicular to the base line. Measure the sides of the triangles so formed and express their ratios as functions of the angle A. Comparing like functions of large and small triangle it will be seen that once an angle is known in degrees, its sine, cosine, etc., are determined irrespective of the length of sides. And, vice versa, if we know the functional values or ratios of certain sides of the right triangle formed about an angle, we have determined the value of the angle in degrees. The Table of Natural Trigonometric Functions, Appendix II, is nothing more than a compilation of these various ratios carefully figured out and placed in the form of a table to assist in the easy solution of problems having to do with the finding of certain parts of a triangle when other parts are given.

With a protractor, measure the angle A of the triangle whose sides were just measured, and compare the ratios of the sides or the functional values with those given in the Table, Appendix III, for the same angle. The larger the scale of the drawing, the greater the accuracy. By making use of the hundredths scale of the framing square together with a finely pointed pair of dividers, variation in values should not be great.