Solve the expression e = sina + cos a + sin2a + cos2a if cos a = -1/4 belongs to (pi, 3pi/2) .

First of all, before calculate sin a, we must establish to
what quadrant belongs. Due to the facts from hypothesis, a is in the interval (pi,
3pi/2), we draw the conclussion that we work in the third quadrant, where the signature
of sin a is minus.

cos a =
-1/4

sin a  = [1- (-1/4)^2]^1/2 (from the fundamental
formula of trigonometry,where sin^2 a + cos^2 a = 1).

sin a
= -(15)^1/2/4

In order to calculate the expression E, first
we have to calculate sin 2a and cos 2a

sin 2a = sin
(a+a)=sina*cosa + sina*cosa=2sina*cosa

cos 2a=
cos(a+a)=cosa*cosa - sina*sina=cos^2a-sin^2a

e= sina a +
cos a+ 2sina*cosa + cos^2a-sin^2a

e= -(15)^1/2/4 - 1/4 +
2*1/4*(15)^1/2/4 + 1/16 - 15/16

After finding the same
denominator, which is 16, we can calculate by grouping the terms which contains (15)^1/2
together and the integer terms together.

e= [-2 -2*(15)^1/2
+ (15)^1/2 - 7]/8=[-9-(15)^1/2]/8