PLANE TRIGONOMETRY. 81 



385. From the equivalence 



log(l +*c- W=T )+ log(l -xi 6 ^) = logfl -x(x + 27^1 8in ^ 



obtain the expansions of cos 2tt0, and sin(2n + l)0 in terms of 

 sin 0. 



386. From the equivalence 



. 0-xJ^T . + 

 -2cos0 + ~ = 4sm -sin- 



resolve the former into its real quadratic factors. 



387. If tan (a + ft J~\) = ^/"^T, a, ft being real, then will a 

 be indeterminate and infinite. 



388. If cos(a-f ^ /s /^T) = cos^ + JS /^T[ sin<, where a, ft, <j> 

 are real ; then will sin < = sin* a. Find also the relation be- 

 tween a and ft. 



389. If tan (a -f ft ^l) = cos <#> 4- sin <, a, ft <#> being 



real, then will 



390. If tan(a + /3 ( y-) = tan ^ + > 8ec ^ A 6 bein S 

 real, then will 



* + 0, 2)3 = log cot ^. 



391. Trove that 



1' 2 f 3 



' = 



_. If in a triangle the sides a, 6, and the angle w-6 

 posite 6 be given, and $ be small ; prove that, approximately 



c atf ab'-Wb-W P 



tt ml +l nr- -E- 



G 



