use the given info. to find the exact value of the trig function.

csc u=-6/5, tan u is greater than 0
Find cos u/2

csc u = -6/5

sinu = -5/6
by Pythagoras, and knowing sin u is negative and tan u is positive
cosu = -√11/6

we know cosA = 1 - 2cos^2(A/2)
or
cosu = 1 - 2cos^2(u/2)
-√11/6 = 1 - 2cos^2(u/2)
2cos^2(u/2) = 1 + √11/6)
cos u/2 = √[(1 + √11/6)/2]

check my arithmetic,

can u simplify the answer any more?????

I had the formula backwards, ...

Should have been

cos A = 2cos^2(A/2) - 1
or
cos u = 2cos^2 (u/2) - 1
-√11/6 + 1 = 2cos^2 (u/2)
-√[1 - √11/6)/2] = cos (u/2)

I picked the negative value, since we knew u was in III, so u/2 was in II, and in II the cosine is negative.

Since you wanted the "exact" value, there is no point simplifying the above answer.

thank u

To find the exact value of the trigonometric function, we need to use the given information. Firstly, we are given that csc(u) = -6/5 and tan(u) is greater than 0.

We can start by finding the value of sin(u) using the reciprocal relationship between csc(u) and sin(u).

csc(u) = 1/sin(u) = -6/5

We can then find sin(u) by taking the reciprocal of csc(u):

sin(u) = 1 / (csc(u)) = 1 / (-6/5) = -5/6

Next, we can find the value of cos(u) using the Pythagorean identity:

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

Plugging in the value of sin(u), we have:

(-5/6)^2 + cos^2(u) = 1

Simplifying, we get:

25/36 + cos^2(u) = 1

Subtracting 25/36 from both sides, we have:

cos^2(u) = 1 - 25/36 = 11/36

To find cos(u), we take the square root of both sides:

cos(u) = ±sqrt(11/36)

Now, since tan(u) is greater than 0 and tan(u) = sin(u) / cos(u), we can determine that both sin(u) and cos(u) must have the same sign (+ or -). Since sin(u) is negative (-5/6), we will take the negative sign for cos(u) to ensure they have the same sign.

cos(u) = -sqrt(11/36)

Finally, to find cos(u/2), we can use the half-angle formula for cosine:

cos(u/2) = sqrt((1 + cos(u)) / 2)

Substituting the value of cos(u):

cos(u/2) = sqrt((1 + -sqrt(11/36)) / 2)

Simplifying further, we have:

cos(u/2) = sqrt((36/36 - sqrt(11/36)) / 2)

cos(u/2) = sqrt((36 - sqrt(11))/72)

Therefore, the exact value of cos(u/2) is sqrt((36 - sqrt(11))/72).