614 



MATHEMATICS. TRIGONOMETRY. 



ANGLES. 



(8). Express each of the trigonometrical ratios of 



an angle A in terms of the sine of A. 

 There are two or three angles, the numerical values of 

 the trigonometrical ratios of which can be easily deter- 

 mined. These angles are 45, 00, and 30. 



6. To find the Trigonometrical Ratioi of an Angle 0/45. 

 A B C, a right-angled triangle. C the right angle. 

 If A - 45, then A - B, and AC - BC. 

 Now, AC + BC-AB. (Eucl. 1.47). 

 /. 2AC 1 -AB, Fig.s. 



or2.BC-AB. 



BC 1 



AB 



oo-^'-AO- 

 and oosin. 45 = sin. (90 45) - sin. 45. 



.'. cos. 46--^ 



Similarly, cot. 45=, 1. 



oosec. 45= J 2. 



1. To find the. Trigonometrical Ratio* of an Angle o/60. 



A B C, an equilateral triangle. The angle A B C is 

 one of 60. Draw A D perpendicular to B C. Now 

 BD- J. BC = J. AB and AD = AB J B D ! = J. 

 AB. 



Fig. 6. 



4 2 



AD_ y3 

 AB~ 2 

 V* 



.'.AD=: 



. f .sin.60 = 



AB. 



tan. 60. 



AB 



~f>T\ 



cotan. ^ = TT) : 



A T> 



cosec. 60= -rv = 



2 



Since 30= 90 60 

 we shall have sin. 30= cos. 60= J. 



And similarly tan. 30= ^ sec. 30= 



cos. 30= 

 Cosec. 60 



Cotan. 30 . 



J. 



8. 



Oeneralitation and Extension of the Principle* and 



Definitions previously laid down. 

 The definitions above given hold good for angles that 

 are less than ninety degrees ; the definition, both of an 

 angle and of the ratios which determine it, admit of and 

 require extension ; the nature of which extension and the 

 principle on which it is made we will now proceed to 

 pxplain. 



9. The vie of the Negative Sign to denote position. 



Let A B be a line, the Fig. 7. 



length of which is a. Let c 



B C be a line, the length of * - ' - '" 



which in 6. Then it is plain that A C is o 6. This 

 distance, A C, is arrived at by measuring a distance (a) 

 to the right from A, and then measuring another dis- 

 tance (6) to the left from li, the + a and the 6 being 

 measured in opposite directions. 



It appears, therefore, that when a stands for a line 

 measured from a given point in one direction, o will 



stand for a line of tho same length measured in the oppo- 

 site direction. In other words, the magnitude of the 

 line is determined by tho number of units in a, while tho 

 direction is determined by its sign. 



It is generally understood that + a signifies a line 

 measured to the right of a given vif. 8. 

 point, as A B, and therefore that., , , 



o signifies a line measure to A 

 the left of the fixed point, as AB' . 



10. Esttnsion of the Definition of an Angle. 



We now proceed to extend the definition of an angle. 

 An angle, as defined by Euclid i. e., as the inclination 

 of one line to another must be less than two right 

 angles. But if we regard an angle as the space swept 

 out by a right line revolving in one plane, about a fixed 

 point in a given straight line, we clearly remove the limit 

 imposed by Euclid's definition on the magnitude of the 

 angle. 



Thus if A be the fixed point K - * 



in the fixed line A B, A P the P 



movable line, let the angle ^^ 



BAP, according to Euclid's 

 definition, be A. 



Now it is plain that in one 

 revolution AP passes through 

 an angle equal to four right 

 angles, or 360. Moreover, A P will always come to its 

 present position after one, two, or any number of revo- 

 lutions ; and therefore, according to our extended defi- 

 nition, BAP may be either A or 360 + A, or 2 X 360 

 + A, or, generally, 360 + A, where n is any integer. 



11. Negative Angle*. 

 In the same manner as we have shown that + a and 



a mean equal lines measured in a contrary direction, 



Fig. 10. 



so + A and 

 A will mean 

 angles measured 

 in contrary di- 

 rections. Thus, 

 A B and A P have 

 the same mean- 

 ing as before, 

 Let A P' be so 

 placed that P'AB 

 = P A B ; then 

 if PAB = A, 

 P'AB A. 



It is plain that if A P' comes into the position A P, it 

 must revolve through an angular space of 360 + A, 

 as denoted by the portion of a dotted circle in figure, or 

 through a certain number of total revolutions besides 

 360 + A. Hence BAP may also be represented by 

 300 + A, - 360 X 2 + A, or generally by 360 x n 

 + A, where n is any positive integer. 



Hence we conclude that if A be any geometrical angle, 

 its most general trigonometrical form will be 



n X 360 + A, 

 where n is any positive or negative integer whatever. 



12. Extension of Definition* of Trigonometrical Ratio*. 



Fig. 11. 



We now pro- 

 ceed to consider 

 the trigonometri- 

 cal ratios of angles 

 greater than a 

 right angle. We 

 shall, in the first 

 instance, confine 

 our attention to 

 B the sines and co- 

 sines of angles, 

 less than four 

 right angles. 



We have al- 

 ready explained 

 that 



