Calculate cos a if sin a = 2/3 and a is in the second quadrant.

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First, we have to emphasize that if a is in the second quadrant, the value of cos a is negative.

Due to the fact that the value of sin a is given, from enunciation, we'll calculate the value of cos a, with the help of fundamental formula of trigonometry:

(sin a)^2 + (cos a)^2 = 1

But sin a = 2/3

(2/3)^2 + (cos a)^2 = 1

We'll subtract (2/3)^2 both sides:

(cos a)^2 = 1 - 4/9

cos a = -sqrt (5/9)

**cos a = -(sqrt 5)/3**

sin a= 2/3

We know that:

sin^2 a + cos^2 a= 1

==> cos^2 a = 1 - sin^2 a

==> cos^2 a = 1- (2/3)^2

==> cos^2 a = 1- 4/9

==> cos^2 a= 5/9

==> cos a = sqrt5/3

But a is in the 2nd quadrant where cos a ia negative:

==> cos a = - sqrt5/3

sina = 2/3 , a is in secon quadrant.

Therefore cos^2 a+sin^2 a = 1.

Or cos a = + or -sqrt(1-sin^2a) = sqrt(1-(2/3)^2) = +or - sqrt(5/9)

= - (sqrt5)/3 as a is in 2nd quadrant the cosa is -ve.

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