If tan a= 12/15, 0 degrees < a < 90 degrees, then find the exact value of each of the following:

A. Sin a/2
B. Cos a/2
C. Tan a/2

Make a sketch using tana = 12/15

then x = 15, y = 12
r^2 = 12^2 + 15^2
r = √369 = 3√41
sina = 12/(3√41) = 4/√41
cosa = 5/√41

Using cos 2A = 1 - 2sin^2 A
or cos a = 1 - 2sin^2 (a/2)
5/√41 = 1 - 2sin^2 (a/2)
2sin^2 (a/2) = 1 - 5/√41
2sin^2 (a/2) = (√41 - 5)/√41
sin^2 (a/2) = (√41-5)/(2√41)
sin (a/2) = √[(√41-5)/(2√41)]

Now use the property: sin^2 x + cos^2 x = 1
to find cos (a/2)
and once you have that, recall that tan(a/2) = sin(a/2) / cos(a/2)

To find the exact values of sin(a/2), cos(a/2), and tan(a/2), we need to use the given information and a suitable trigonometric identity.

Let's start with the given value of tan(a) = 12/15. This means that the opposite side to angle a is 12 units, and the adjacent side is 15 units. We can use the Pythagorean theorem to find the hypotenuse of the right triangle formed by these sides.

Using the Pythagorean theorem:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 12^2 + 15^2
hypotenuse^2 = 144 + 225
hypotenuse^2 = 369
hypotenuse = sqrt(369)

Now, we have all the side lengths of the right triangle formed by angle a, i.e., the opposite side (12), adjacent side (15), and hypotenuse (sqrt(369)).

To find sin(a/2), we can use the half-angle formula for sine, which states that:
sin(a/2) = sqrt((1 - cos(a)) / 2)

To find cos(a/2), we can use the half-angle formula for cosine, which states that:
cos(a/2) = sqrt((1 + cos(a)) / 2)

To find tan(a/2), we can use the half-angle formula for tangent, which states that:
tan(a/2) = sin(a) / (1 + cos(a))

Now, let's plug in the values we have to find the exact values:

A. Sin(a/2):
First, we need to find cos(a) using the given information:
cos(a) = adjacent / hypotenuse
cos(a) = 15 / sqrt(369)

Now, we can find sin(a/2):
sin(a/2) = sqrt((1 - cos(a)) / 2)
sin(a/2) = sqrt((1 - 15 / sqrt(369)) / 2)

B. Cos(a/2):
cos(a/2) = sqrt((1 + cos(a)) / 2)
cos(a/2) = sqrt((1 + 15 / sqrt(369)) / 2)

C. Tan(a/2):
First, we need to find sin(a) using the given information:
sin(a) = opposite / hypotenuse
sin(a) = 12 / sqrt(369)

Now, we can find tan(a/2):
tan(a/2) = sin(a) / (1 + cos(a))
tan(a/2) = (12 / sqrt(369)) / (1 + 15 / sqrt(369))

These are the exact values of sin(a/2), cos(a/2), and tan(a/2) using the given information. You can further simplify them if needed by rationalizing the denominators or using special right triangles.