1. CosA = 3/5

To find the value of sin 2A, we need to use trigonometric identities. One of the most commonly used identities is the double-angle identity for sine, which states:

sin 2A = 2 sin A cos A

Given that cos A = 3/5, we can find sin A using the Pythagorean identity, which states:

sin^2 A + cos^2 A = 1

Substituting the value of cos A, we have:

sin^2 A + (3/5)^2 = 1
sin^2 A + 9/25 = 1
sin^2 A = 1 - 9/25
sin^2 A = 16/25

Taking the square root of both sides, we get:

sin A = ±4/5

Since we are given that sin A is positive, we have sin A = 4/5.

Now, substituting the values of sin A and cos A into the double-angle identity for sine, we have:

sin 2A = 2(4/5)(3/5)
sin 2A = 24/25

Therefore, the value of sin 2A is 24/25.

2. SinA = 12/13

Similar to the previous case, we can use the double-angle identity for sine:

sin 2A = 2 sin A cos A

Given that sin A = 12/13, we need to find the value of cos A. Using the Pythagorean identity:

sin^2 A + cos^2 A = 1

Substituting the value of sin A, we have:

(12/13)^2 + cos^2 A = 1
144/169 + cos^2 A = 1
cos^2 A = 1 - 144/169
cos^2 A = 25/169

Taking the square root of both sides, we get:

cos A = ±5/13

Since we are given that cos A is positive, we have cos A = 5/13.

Now, substituting the values of sin A and cos A into the double-angle identity for sine, we have:

sin 2A = 2(12/13)(5/13)
sin 2A = 120/169

Therefore, the value of sin 2A is 120/169.

3. TanA = 16/63

To find the value of sin 2A, we need to use the double-angle identity for sine:

sin 2A = 2 sin A cos A

Given that tan A = 16/63, we can find the values of sin A and cos A using the definitions of tangent:

tan A = sin A / cos A

Substituting the value of tan A, we have:

16/63 = sin A / cos A
16 cos A = 63 sin A

Using the Pythagorean identity:

sin^2 A + cos^2 A = 1

Substituting the value of cos A, we have:

sin^2 A + (63/16)^2 sin^2 A = 1
(1 + (63/16)^2) sin^2 A = 1
(1 +