NOTE. In algebraic work of this sort the student will find it convenient, instead of writing sin a, cos a, etc., in full, to use a single symbol for each function; for example, he may put: It will be seen that the second members of most of these equations are surds, showing that their values may be either positive or negative. The result may be expressed thus: Whenever one trigonometric function is given, the four other functions which are not its reciprocal may have either of two equal values with contrary signs. This arises from the fact that every such function may belong to either of two angles, and affords an example of the correspondence between geometric and algebraic results. EXERCISES. 1. From the special values of sin 30° and sin 45° found in § 30, namely, sin 30° = 1, sin 45° = √, find the values of the other five functions of 30° and of 45°. cosec 30° = 2; 2. It has been shown that the three products sin x cosec, cosec 45° = √2. tan X cot, and cos X sec are each unity. y When we replace the functions by the lines which represent them, the products represent the areas of the rectangles contained by the lines, and unity M T N X Р is replaced by the square of the radius. Hence we have the following theorems, which are to be proved by the similar triangles in the accompanying figure, where the construction is that of the trigonometric functions. 66 I. Rectangle XT. YN = OX', corresp. to tan X cot = 1. II. Rectangle MP.ON = 0X', III. Rectangle OP.OT = 0X', "sin X cosec = 1. 66 cos X sec = 1. 3. From the value of sin 18° in § 30 find the sides of the regu lar inscribed and circumscribed decagons of a circle of radius a. 10. What angle is that of which the tangent is double the sine? 11. What must be the value of the cosine in order that the tan gent may be n times the sine? Ans. 1 n 12. Prove (r cos x) + (r sin x sin u) + (r sin x cos u)2 = r2. 13. Prove (a sin y)2 + (a cos y sin d)2 + (a cos y cos d')2= a3. 14. Prove (cos a cos b- sin a sin b)2 +(sin a cos b + cos a sin b)2 = 1. 15. Of what angle is the secant double the sine? 16. Of what angle is the secant four times the cosine? CHAPTER III. OF RIGHT TRIANGLES. 35. Fundamental relations. Let OCN be a right triangle of which a and b are the sides which contain the right angle, c the hypothenuse, a and ẞ the angles opposite a and b respectively. If we take ON as a radius and draw the arc NX from the 'centre O, the side NC will, by definition, represent the sine of XON, and OC its cosine, when the radius is ON. That is, NO = sin a; ୦୯ ON = cos α. (1) We may show in the same way, by taking N as the centre and NO as the radius, ос ON = sin ß; NC = cos B. (2) We might also have deduced these equations from (1), because B = 90° α, whence Putting NC = a, OC = b, ON = c, the equations (1), (2), and (3) give the relations We may express the same relations in terms of ß, using the complementary functions, as follows: These relations may be summed up in the following general theorems: I. The hypothenuse of any right triangle is equal to a side into. the secant of its adjacent angle or the cosecant of its opposite angle. II. A side is equal to the hypothenuse into the sine of the opposite angle or the cosine of the adjacent angle. III. One side is equal to the other side into the tangent of the angle adjacent to that other side or the cotangent of the angle adjacent to itself. EXERCISE. Show by the above equations how each side will be expressed. in terms of the others when a 30° and when a = 45°, using the, values of sin a, etc., already found—namely, and show how the results agree with those of elementary geometry. 36. Examples and exercises in expression. In the accompanying figure OQN, ONP, and OXN |