Math trigonometry transformation

Question

Hi,

I haven't done math in a while, and stumbled upon this thing. The angle ($\arccos 7/25) is given, and i have to calculate the cosine of it's half. I've used the basic formula for cosine of an half angle, and replaced it.

I'm pretty sure the answer i've got is incorrect. But still don't get it, where i've made the mistake.

Thank you in advance.

Answer 1

It's almost OK. You lack some rigour: you should write $\,\cos\dfrac\theta2=\pm\sqrt{\dfrac{1+\cos\theta}2}$, and explain why you choose +.

What is wrong is fractions:

$$\sqrt{\dfrac{\cfrac{32}{25}}{2}}=\sqrt{\dfrac{16}{25}}=\dfrac 45.$$

Answer 2

HINT: use this formula: $$\cos(\theta/2)=\pm\sqrt{\frac{\cos(\theta)+1}{2}}$$

Answer 3

Note that: arccos(7/25) lies in either 1st quadrant or fourth quadrant. (cos is +ve), so 2*arccos(7/25) lies in 2nd quadrant and 3rd quadrant, where cos is negative. So the answer should be -4/5.(by dividing the numerator and denominator by $sqrt(2)$)

Other than this, the answer is correct.