
Re: Proving trigonometric identities
Posted:
Jan 21, 2009 11:37 PM


On Jan 22, 3:32 am, Albert <albert.xtheunkno...@gmail.com> wrote: > Moving away just a little away from proof of trigonometric identities > and onto solving 'simple' trigonometric equations: > > When I get questions like: > Find solutions in the range 0 degrees <= x <= 360 degrees for these > trigonemetric equations: > (a) sin x = 1/2; > (b) ... > ... > > Do I figure out one answer (say 30 degrees) and then think: there > might be another answer and if so, the sin of it better be positive as > well, which would mean it'd be in the 2nd quadrant and figure out 150 > degrees? Is that the way to do these problems because I don't have any > worked examples whatsoever. > > I don't have any worked examples for questions like these either: > (a) Write sin(theta) and tan(theta) in terms of cos(theta) when theta > is in the first quadrant > (b) If cosA = 9 / 41, and A is in the first quadrant, find tanA and > cosecA. > > I know that you can get a calculator and solve for A using inverse > cosine immediately, but what is the real intention of this question? > The trig identity tan(theta) = sin(theta) / cos(theta) can be re > arranged for finding a value for sin(theta) but i) what is the > significance of which quadrant theta is in (and) how do I find tanA in > part b?
In general, the equation cos(A) = k for some known k (such as 9/41 in your case) has two solutions for A in the range 0 to 2*pi (i.e. 0 to 360 degrees). Call these solutions A1 and A2. While by definition cos (A1) = cos(A2), this equality is not generally true for the other trig functions. For example, it's not generally the case that tan(A1) = tan (A2) (in fact, tan(A1) = tan(A2)). So, if you only know that cos(A) = k, then tan(A), for example, is ambiguous. To know which value is intended it's necessary to have further information about A, such as the quadrant in which it lies.

