# Calculate the expression E = sin a + cos a + sin 2a + cos 2a if cos a = -1/4 and belongs to ( pi , 3pi/2 )

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E= sin a +cos a + sin2a+ cos 2a

cos a =-1/4

==> sin a= sqrt(1-1/16)= sqrt(15)/4

we know that sin 2a= 2sin(a)*cos(a)

= [2sqrt(15)/4](-1/4)=-sqrt(15)/8

cos 2a= cos^2(a)-sin^2(a)

= 1/16 -15/16= -14/16= -7/8

Now let us substitute:

E= -sqrt(15)/4 -1/4 -sqrt(15)/8 -7/8

= -3sqrt(15)/8 -9/8

E = sina+cosa+sin2a+cos2a , if = -1/4.

To calculate a.

Solution:

cosa = -1/4 a belongs to (pi to 3pi/2)

Therefore, sina = -sqrt[1-(-1/4)^2 ] = -sqrt (15/16) , as a belomgs to (pi ,3 pi/2), which is the 3rd quadrant where both sine cosine ratios are -ve.

We know that sin2a = 2sinacosa = 2[-sqrt(15/16)](-1/4) = 1/8(sqrt 15) andos2a = 2cos^2-1 = 2(-1/4)^2 -1 = 2/16-1 = -14/16 = -7/8. Substituting in the expression E ,we get:

So E = -sqrt(15)/4+(-1/4)+sqrt15/8 -7/8

= -sqrt15/8 + - 9/8= -sqrt15/4 -1 = -(sqrt15+9)/8 = --1.6091222918.

First of all , before calculate sin a , we must establish in which quadrant is the angle located. Due to the facts from enunciation, a is in the interval (pi, 3pi/2), so we conclude that we'll work in the third quadrant, where sin a is negative.

cos a = - 1/4

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

sin a = - sqrt(15)/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 = - sqrt(15)/4 - 1/4 + 2*1/4*sqrt(15)/4 + 1/16 - 15/16

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

**E= (-2 - 2*sqrt 15 + sqrt15 - 7)/8 = (- 9 - sqrt15)/8**