Plane trigonometry, by S.L. Loney.

[EXS. XXI.] TRIGONOMETRICAL EQUATIONS.

143

11. cos (30 + a) cos (30- a) + cos (50 + a) cos (50 - a) = cos 2a.
12. cos?-=cos(-2) 0 + sin0. 13. sin 0=sin 0 +sin 0.
14. sin mO + sin n = 0. 15. cos mO + cos n = 0.
16. sin2 n - sin2(n -1)0= sin2. 17. sin 30 + cos 20 = 0.
18. V3cos 0+sin 0=/2. 19. sin 0+cos 0=/2.
20. /3 sin 0 - cos 0 = 2. 21. sin x + cos x = /2 cos A.
22. 5 sin 0 + 2 cos 0 = 5 (given tan 21~ 48' = 4).
23. 6 cos x + 8 sin x = 9 (given tan 53~ 8'= 1 ).
24. 1 + sin2 0 = 3 sin 0 cos 0 (given tan 71~ 34'= 3).
25. cosec0= cot 0 +,/3. 26. cosec x = 1 + cot x.
27. (2+V/3)cos0=l-sin. 28. tan 0+sec o=^/3.
29. cos 20 = cos2 0. 30. 4 cos 0 -3 sec 0=tan 0.
31. cos 20 +3cos =0. 32. cos 30+2cos0=0.
33. cos20=(/2+1) (cos0 — )
34. cot -tan 0=2. 35. 4 cot 20= cot20-tan20.
36. 3 tan (0 - 15~)=tan (0 + 15~).
37. tan 0 + tan 20 + tan 30 =0.
38. tan 0 + tan 20 + 3 tan 0 tan 20 =^3.
39. sin 3a = 4 sin a sin (x + a) sin (x -a).
40. Prove that the equation x3 - 2x + 1 = 0 is satisfied by putting for x
either of the values
V/2 sin 45~, 2 sin 18~, and 2 sin 234~.
132. Ex. To trace the changes in the sign and
magnitude of the expression sin 0 + cos 0 as 0 increases from
0 to 360~.
We have sin 0 + cos 0 =/2 2 sin 0 + -2 cos ]
= 12 [sin 0 cos 45~ + cos 0 sin 45~] = 2 sin (0 + 45~).