Plane trigonometry, by S.L. Loney.

84 TRIGONOMETRY. [Exs. XI.]
22. Find the angles between 0~ and 360~ which have respectively (1)
their sines equal to /2 (2) their cosines equal to -' and (3) their tangents equal to,.
23. Taking into consideration only angles less than 180~, how many
5 1 4
values of x are there if (1) sin x=7, (2) cos= 5, (3) cosx= - 5, (4)
2
tanx. =3, and (5) cotx= -7?
24. Given the angle x construct the angle y if (1) siny =2 sin x, 2)
tan y = 3 tan x, (3) cos y = 2 cos x, and (4) sec y = cosec x.
25. Shew that the same angles are indicated by the two following
formulae: (1) (2n- 1) + (- 1)n 3, and (2) 2n7r 4-, n being any integer.
26. Prove that the two formulae
(1) (2n+ )7r a and (2) n7r+(-l1) (2-a
denote the same angles, n being any integer.
Illustrate by a figure.
27. If 0-a=nrr+(-1)-j3 prove that 0=2mnz7r+a +J or else that
=(2m+l)7r + a - where mn and n are any integers.
28. If cospO + cos q=O0, prove that the different values of 0 form two
27w
arithmetical progressions in which the common differences are - and,
P+q
27r
-- respectively.
P-q
29. Construct the angle whose sine is 2 +/-5
86. An equation involving the trigonometrical ratios
of an unknown angle is called a trigonometrical equation.
The equation is not completely solved unless we
obtain an expression for all the angles which satisfy it.
Some elementary types of equations are solved in the
following article.