Use the given information to find the exact function value. Simplify your answer as much as possible. cosa = 4/5, 3π/2<a<2π. What is sin(a/2), cos(a/2), and tan(a/2)?

Answer 1

Hi Abbi,

Hopefully I can help you with the first part, finding the exact function value of sin(a/2)

We are given that the cos a = 4/5 and that the angle a is between 3π/2 and 2π.

So, what do we know about the angle a/2? a/2 must between 3π/4 and π. ( please check if this makes sense)

We have the double angle formulas ( If these are tricky to memorize, you can derive them as well)

sin(a/2) = ± √[(1 - cos a)/2]

cos(a/2) = ± √[(1 + cos a)/2]

Please note that we have a plus or minus involved, as this comes into play later.

Since we know cos a = 4/5, plugging in we get

sin(a/2) = ±√[(1 - 4/5)/2]

sin(a/2) = ±√[(1/5)/2]

sin(a/2) = ±√(2/5)

We are given that sin(a/2) is either √(2/5) or -√(2/5. The question asks for the exact function value, so we need to pick the right value.

If we know a/2 is between 3π/4 and π, or in the 2nd quadrant. (Please check that you agree with this)

Since sine is positive in the second quadrant, we know that the answer has to be sin(a/2) = √(2/5).

Hopefully you can figure out cos(a/2) and then tan(a/2) now, if not, please let me know if any other question.