what is the exact value of tan(arccos((x-1)/8)) after you get to sqrt((x+7)(-x+9))/(x-1)
Huh? That is the exact value.
That's what I got
or
you could have left it as
√(63 +2x - x^2)/(x-1)
test it with your calculator , by picking something like x = 3.5, make sure your calc is in radians
(worked for me)
To find the exact value of tan(arccos((x-1)/8)), we can use the following steps:
Step 1: Start by using the definition of inverse cosine (arccos) to find the angle whose cosine value is ((x-1)/8).
arccos((x-1)/8) = θ, where cos(θ) = ((x-1)/8)
Step 2: Since we know that cos(θ) = ((x-1)/8), we can draw a right triangle with the adjacent side as (x-1) and the hypotenuse as 8. This triangle will help us find the remaining side (the opposite side).
|
(θ) | *
| \
| ((x-1)/8)
|_________\
8
By using the Pythagorean theorem, we can solve for the length of the opposite side:
opposite side = sqrt(8^2 - (x-1)^2)
= sqrt(64 - (x^2 - 2x + 1))
= sqrt(65 - x^2 + 2x)
So, the length of the opposite side is √(65 - x^2 + 2x).
Step 3: Now we have all the sides of the right triangle. To find the tangent of the angle θ, we use the ratio of the opposite side over the adjacent side.
tan(θ) = (opposite side) / (adjacent side)
= √(65 - x^2 + 2x) / (x-1)
Therefore, the exact value of tan(arccos((x-1)/8)) = √(65 - x^2 + 2x) / (x-1).