## Tuesday, April 7, 2015

### [AM_FTDA20150402] Proof of Triple Angle Sine Identity

Question

Solution

Discussion
We are given a triple angle, and we want an expression in terms of  sin q  only, without any double or triple angle.  First, we split up 3q  into  2q +q.  see [1].  This allows us to use the compound angle formula, and then double angle formulas.  At every step, it is a good tactic is to compare what you have with what you want.  This problem solving heuristic is not in the official list, but from my experience it is a useful one.  So at [2], we have three choices for the cos 2q :  cos 2q  = cos2q  – sin2q ,  cos 2q  = 2cos2q  – 1,  and  cos 2q  = 1– 2sin2q.  Which one shall we choose?  Since everything needs to be expressed in terms of  sin q  only,  the best choice is the third formula.  At [3], we have a  cos2q  appearing in the first term.  Again we express that in terms of  sin q  via  cos2q  = 1 – sin2q.  After that, we simplify to get to the RHS.

H10. Simplify the problem
H11. Solve part of the problem
Hxx. compare what you have with what you want

Suitable Levels
* revision for GCE ‘A’ Level H2 Mathematics
* revision for IB Mathematics HL / SL
* other syllabuses that teach further trigonometry