Date: Jan 21, 2009 11:37 PM
Author: matt271829-news@yahoo.co.uk
Subject: Re: Proving trigonometric identities
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.