# Given that sin a = 3/5 and a is in the interval (90,180), calculate tan a.

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sina = 3/5

We know that :

sin^2 + cos^2 = 1

==> cosa = sqrt(1-sin^2a)

= sqrt(1-9/25) = sqrt(16/25) = 4/5

==> cosa = 4/5

Since a is in the second quadrant , then sin>0 and cos < 0

==> cosa = -4/5

==> tana = sina/cosa = (3/5) / (-4/5)= -3/4

**=> tan(a) = -3/4**

For an angle x that lies in the interval 90 degrees to 180 degrees, sin x has positive values , cos x on the other hand in negative.

We have sin a = 3 / 5. We also know that sin^a+ cos^a =1.

Therefore we can get cos a = sqrt ( 1- sin^a)

= sqrt [ 1- (3/5 )^2]

= sqrt [ 1- ( 9/ 25)]

= sqrt [ (25 -9)/ 25]

=sqrt (16/ 25)

= - 4/5

Therefore tan a = sin a / cos a = (3/5)/ (-4/5) = 3/ -4 = -3/4.

**The value of tan a is -3/4**

sina = 3/5

To calculate for the value of tan a in the interval (90, 180).

In the seond quadrant sina is positive, cosa is negative. So the value of tana is negative.

sina = 3/5...............................................(1)

cos a = - (1-sin^2a)^(1/2).

cosa = -{1-(3/5)^2}^(1/2)

cosa = - (25-9)^(1/2)/5

cosa = -{16^(1/2) }/5

cosa = -4/45..............................................(2)

From (1) and (2), we get tana.

tana = sina/cosa = (3/5)/(-4/5)

tana = -3/4.

If the angle a is in the interval (90,180), that means that the angle is located in the 2nd quadrant, then the value of tangent function is negative.

The tangent function is the ratio:

tan a = sin a/ cos a

Since the value of cos a is negative in the second quadrant, that means that the value of the ratio is negative, too.

We'll calculate cos a from the fundamental formula of trigonometry:

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

9/25 + (cos a)^2 = 1

We'll subtract 9/25 both sides:

(cos a)^2 = 1 - 9/25

(cos a)^2 = (25-9)/25

(cos a)^2 = 16/25

cos a = -4/5

We'll choose only the negative value for cos a, since a is in the 2nd quadrant.

tan a = -(3/5)*(5/4)

We'll reduce the like terms:

**tan a = -3/4**