Plane trigonometry with practical applications, by Leonard E. Dickson.

Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 167
cos A = + ll-cX2, cos B = -+ /- y2,
sin (A + B) = sin A cos B + cos A sin B = x1_- y2 + y-1- x2.
EXERCISES ON INVERSE FUNCTIONS
1. From the graphs (Figs. 88-90) of the tangent, cotangent, cosecant, and secant, derive the graphs of their inverse functions and justify
the definition of their principal values.
Find the values of
2. sin (Arc tan 1). 3. sec (Arc tan 2). 4. cos (Arc cos 2).
5. tan (2 Cos-1 ). 6. cos (2 tan-1 ~3). 7. tan (Sin1 5).
Prove the following formulas (with x and y numerically <1 in Exs. 8,
10, 12, 14 and 15):
2c
8. cos (2 cos1 x) = 2 x2-1. 9. tan (2 tan c) = 2
10. Arc cos x= Arc cos (2 x2-1) if 0 < x < 1.
2n7
11. Arc sin i +Arc sin 1 = 2 -12. Sinl x + Cosl x = 7r.
13. Sin- 1 - Sin-1 2 = Cos-1. _
14. cos (Sin1 x - Sin'1 y) = xy + 1-x2. V1 -y2.
15. sin (Sin' x - Sin-1 y) = xW/1-y2 - y/1l-x2.
16. Tan'1 x- Tan'1 y = Tan-l -y if x>0, y>0.
1+xy
17. 2 Tan-1 2 = Tan-1 2 18. 3 Sin-1 = Sin1
— i. 1 —C19. 'Sin-1 3 + Sin-l 8 = Sin-1 7 7 20. Cos- Cos-1 2 = Cos'1.
21. Arc tan + Arc tan = 22. Tan'-1 +Tan- =
23. 2 Tan' + Tan'l =- 24. Tan' + Tan- 2=
4~ —~a 2
25.* Tan'1 x + Tan'1 y =n7r + Tan- 1 -xy where n = 1 if x and y are
l -xy
positive and xy > 1; n -1 if x and y are negative and xy > 1; n = 0
in all remaining cases.
26.* Arc tan 5 + Arc tan (-3)= Arc tan i.
Solve and discuss the two equations:
27.* cos-1 x + cos'- (1-x) = cos-' (-x).
28.* tan-L (x + 1) + tan'l (x — 1) = tan'1 s3.