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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