sin(2sec^-1 u/2)

Question

There is a different problem using U substitution that I have been trying to solve for the majority of the day. If someone could walk me through this it would be great. Thank you. 

Answer 1

sec(θ) = hypotenuse/adjacent = h / a

 

sec^-1(u/2) is and angle, so let sec^-1(u/2) = θ.

 

Let u = h and 2 = a.

 

Then, o = √(u² - 4), the opposite side, by the Pythagorean theorem.

 

sin(2θ) = 2sin(θ)cos(θ) = 2(o/h)(a/h) = 2(√(u² - 4)/u)(2/u)

 

= 4√(u² - 4) / u²

 

 

So, sin(2sec^-1(u/2)) = 4√(u² - 4)/u² with u ≥ 2.

Answer 2

You want the sine of twice the angle whose secant is u/2.

Let x be this angle.  If secx=u/2 then cosx=2/u and sinx=√(1-4/u²)=(√(u²-4))/u

sin2x=2sinxcosx=(4√(u²-4))/u²