Clo 



MATHEMATIC& TRIGONOMETRY. 



[TRIflONOMmilCAI. RATIOS. 



Now 



Un. 180' - 



tin. 180 0_ 

 ooa. 180"" 1 ' 



- "- 





If we apply similar reasoning to the various trigono- 

 metric functions to that employed in discussing the 

 variations of the sine of A, we obtain results which may 

 be arranged in a tabular form, as follows : 



QUADRANT. 



The student will do well to verify carefully all the 

 results given in this table ; he will also observe that the 

 trigonometrical ratios illustrate the principle that if a 

 function of a variable changes its sign, it must pass 

 through the values of either zero (.0) or infinity (t/>). 



As the values of the ratios are continuous, the ratios 

 increase gradually to their greatest value, and then de- 

 crease to their least Thus, to take the case of the sine 

 of an angle, which we call 0. 



Sine increases from 0, when = up to 1, when 

 = 90. It then decreases to 0, when = 180" ; after 

 which it still further decreases till it equals 1, when 

 6 = 270, and finally increases up to 0, when = 300. 



15. To determine all the atigles which have the same sine, 

 or cosine. 



There is another class of questions presented to us by 

 this extension of our definition of an angle, viz., having 

 given a trigonometrical ratio of an angle, to find all the 

 angles corresponding to it. 



For example, tan. = p. 



Now, if we did not reckon any angles but those less 

 than 180, as is the case in geometry, we could only have 

 one value of corresponding to a given value of tan. 9. 

 Suppose this value = a. 



Then, if we take the trigonometrical or generalised 

 conception of an angle, we shall have another = 180 



+ 



And since no trigonometrical ratio changes either its 

 value or its sign when its angle is increased by any 

 multiple of 360, it is plain that in addition to a we shall 

 have a series of values, 360 -f a, 2 x 360 + a, 3 X 300 



-f- a + n 360 -f- a ; and in addition to the value 



180 -f a, we shall have a series of values, 360 + 180 

 + a, 2 X 360" + 180 + o, 3 X 360 + 180+ a, .... n 

 360 9 + 180 + a. Both these series may be included in 

 one formula, 



= m 180 + a, 



where m is any integer number whatever. 

 In the same manner, if 



Cos. - q, 



and a be the value of less than 180, which has for its 

 cosine q, then all the values of which have a cosine q 

 are included in the formula, 



- m. 360 a, 



whore m is any integer. 

 And similarly if 



Sin. q, 



the value of is included in the formulas 



- 2m 180 + o, 

 and0- (2m + 1)180 -a, 



being any integer whatever. 



These may be included in one formula, as follows : 



where k is any integer whatever ; for when k is even, 

 ( 1)* is positive ; and when k is uneven, (1)* is 

 negative. 



As practice in the preceding articles, the student may 

 verify the following results : 



1. Sin. 180 = 0. 



2. Sin. 135 = }., 



3. Cos. 135= 4- 2 = cos. 225. 



4. Cos. 315= , s = sin. 405. 



6. Sin. 120= -4- =-sin. 300. 



f 



6. Cos. 300 = 



7. Sin. 150= 



8. Cos. 300= 



'-COS. 150. 



Given that sine = -=-, show that the following are the 



values of which satisfy that equation, 30 150, 390, 

 510, 750, 870, etc. 



Given that cos. = ,-j< show that the following are the 



values of which satisfy this equation, 45, 315, 405, 

 675, 765, <bc. 



Given that tan. = 1, show that the values of are 

 45, 225, 405, 585, 765, <bc. 



THE RELATIONS BETWEEN THE TRIGONOMETRICAL FUNC- 

 TIONS OF DIFFERENT ANGLES. 



The formulas we have already proved hold good of the 

 ratios of the same angle ; we now proceed to investigate 

 the formulas which express the relations between two or 

 more diflerent angles. There is a very great variety of 

 formulas of this kind, and they admit of an almost in- 

 finite number of combinations and modifications. They 

 are, however, all derived mediately or immediately from 

 the following four : 



Sin. A + B) = sin. A, cos. B + sin. B, cos. A (8). 



Sin. A B) = sin. A, cos. B sin. B, cos. A (!)). 



Cos. A + B) = cos. A, cos. B sin. A, sin. B. . . , 

 Cos. A B) cos. A, cos. B + sin. A, sin. B. . . , 



These four formulas can be easily remembered and 

 it is of great importance that they should be remembered 



