Plane trigonometry, by S.L. Loney.

126

TRIGONOMETRY.

[Exs. XVIII.]

A
32. If sin - be determined from the equation
A A
sin A sin - 3 sin
3 2'
prove that we should expect to obtain also the values of
7r-A +. 7A
sin and - sin
3 3
A
33. If cos - be found from the equation
A A
cos A = 4 cos3 3 cos
3 3
prove that we should expect to obtain also the values of
27r-A 27r+A
cos - and cos --
3 3
120. By the use of the formulae of the present
chapter we can now find the trigonometrical ratios of
some important angles.
To find the trigonometrical functions of an angle of 18~.
Let 0 stand for 18~, so that 20 is 36~ and 30 is 54~.
Hence 20 = 90~ - 30,
and therefore
sin 20 = sin (90 - 30)= cos 30.. 2 sin 0 cos 0 = 4 cos3 0 - 3 cos 0 (Arts. 105 and 107).
Hence, either cos 0 = 0, which gives 0 = 90~, or
2 sin 0 = 4 cos2 0 - 3 = 1 - 4 sin2 0..'. 4 sin2 0 + 2 sin 0= 1.
By solving this quadratic equation, we have
+ /5 - 1
sin0 -