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

On Jan 22, 3:32 am, Albert <> 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.