36 PRACTICAL MATHEMATICS 



Using relation (8), cos (x + h) cos x = 2 sin ( x + - J sin - 



These results will be used in the differentiation, from first 

 principles, of sine and cosine functions. 



(b) If A = pt and ~B = pt - c 



Then A + B - 2pt - c and A - B = c 



From (5) sin pt cos (pt c) = - {sin (2pt c) + sin c } 



M 



From (6) cos pt sin (pt c) = - {sin (2pt c) sin c } 

 From (7) cospt cos (pt c) = o( cos ( 2 P^ ~~ c ) + cos c ) 

 From (8) sin j> sin (pt c) = - {cos c cos (2pt c)} 



This method of transformation, and these results, will be found 

 to be very useful in our subsequent work on Fourier's Series, and 

 also to find the mean values of periodic functions. 



EXAMPLES II 

 Solve the following triangles : 



(1) (a) 3 sides 18, 14, 9. 



(b) 3 sides 7-36, 5-72, 3-84. 



(2) (a) Sides 22 and 31, included angle 62. 

 (b) Sides 5-16 and 3-96, included angle 55. 



(3) (a) Sides 9 and 8, angle 56 opposite to the smaller side. 

 (b) Sides 3-72 and 2-25, angle 32 opposite to the smaller 



side. 



(4) (a) Sides 17 and 20, angle 38 opposite to the larger side. 

 (&) Sides 3-92 and 5-72, angle 44 opposite to the larger 



side. 



(5) (a) Base 17, angles at the base 29 and 44. 

 (b) Base 2-96, angles at the base 34 and 61. 



(6) The sides of a triangle are 5-6, 4-4, and 2-8. Find the area, 

 the angles, and the lengths of the perpendiculars. 



(7) ABC is a triangle, right-angled at C ; the angle ABC 

 is 75 ; the side AC is divided into four equal parts by points 

 D, E, and F. Find the angles DEC, EEC, and FBC. 



(8) ABCD is a quadrilateral. AB = 1-8", BD = 2-4", DC = 3-4", 

 DA = 2-6", and BC = 3-2". Find the area, the angles, and the 

 length of the diagonal AD. 



(9) Write down the values of sin 215, cos 93, tan 321, cos 

 236, sin 112, tan 184, sin 527, cos 412, tan 729, sin ( - 312), 

 cos ( - 196), and tan ( - 521). 



