Question

Velocity/tangent vector example #2 (see link below): how do we know what trigonometric identities to apply? The example seemingly arbitrarily chooses to re-express everything in terms of the half-angle. On a normal problem, is there any indication as to what identities should be used? Or alternatively, is there a solution with only simple identities (I tried but couldn't simplify it enough)? http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-c-parametric-equations-for-curves/session-20-velocity-and-arc-length/MIT18_02SC_we_11_comb.pdf

Answer 1

You could leave the answer in terms of theta rather than theta/2
However, when you integrate to find the arc length I believe you will want to do a trig substitution. At that point you might go back and simplify your previous expressions using that same substitution...