26 PRACTICAL MATHEMATICS 



If an angle is such that the movable arm takes up a position in 

 a certain quadrant, then the trigonometrical ratios of that angle 

 must be given the algebraical signs peculiar to that quadrant. 



To find sin 123 32', cos 209 19', and tan 324 43'. 



(a) 123 32' is an angle in the second quadrant. 



Then sin 123 32' is + 

 But 123 32' = 180 - a 



a = 56 28' 

 Hence sin 123 32' = + sin 56 28' = + 0-8335 



(b) 209 19' is an angle in the third quadrant. 



Then cos 209 19' is - 

 But 209 19' = 180 + a 



a = 29 19' 

 Hence cos 209 19' = - cos 29 19' = - 0-8720 



(c) 324 43' is an angle in the fourth quadrant. 



Then tan 324 43' is - 

 But 324 43' = 360 - a 



a = 35 17' 

 Hence tan 324 43' = - tan 35 17' = - 0-7072 



13. Angles greater than 360. The rotation of the moving arm 

 with reference to the fixed arm is not limited to one complete 

 revolution : each revolution increases the magnitude of the angle 

 by 360 : but for a certain angle the moving arm is sure to take 

 up a position in one of the four quadrants. Thus an angle greater 

 than 360 can be taken to be made up of two parts : an exact 

 multiple of 360, and an angle between and 360 ; and it is 

 the second part which must be used in order to determine the 

 trigonometrical ratios. 



To find cos 829. 



829 = 2 x 360 + 109 



Then cos 829 = cos 109 



But 109 is in the second quadrant 



Then cos 109 is - 



But 109 = 180 - a 



a= 71 

 Then cos 829 = cos 109 = - cos 71 = - 0-3256 



14. Negative singles. Up to now we have taken the moving 

 arm as rotating in anti-clockwise direction, but that arm can 

 also rotate in the opposite direction that is, in clockwise direc- 

 tion. In order to distinguish between these two directions we 

 take an angle measured in clockwise direction as negative. Now 

 an angle of ( a) represents an angle a measured in clockwise 

 direction, and the moving arm takes up a certain position in one 

 of the four quadrants ; but that same position could be obtained 



