Suppose − π/2 < φ < 0 and cos ⁡ ( φ ) = 2/3. What is cos ⁡ ( φ/2 )?

Question

Need to know if the answer will be positive or negative pls!

Answer 1

If -π/2 <x <0, -π/4 < x < 0. This is a subset of the original interval, where cos is positive, so cos is positive for x/2; you could also say that all values of the subset are in Quadrant 4, where cos is positive. You may have gotten concerned because you were thinking about whether you should be taking half of values between 3π/2 and 2π, which have the same values for the six trig functions. However, the question asks about x/2, and x is simply a number, so normal rules apply for taking half of it.

It sounds like you knew how to calculate the half-angle formula. cos(θ/2) = √((1+cos(θ))/2) =

√((1+2/3)/2) = √(5/6) If you don't like square roots in the denominator, this equals √30 / 6