Solve for x

Date: 10/26/2001 at 00:02:26
From: veronica
Subject: Trig equations

Please help me solve for x:

Solve over (0,2pi)

1. tan^2 x = 2sec x - 1    2. sin x - sqrt3 cos x = 1 

Thank you.

Date: 10/26/2001 at 11:40:13
From: Doctor Rob
Subject: Re: Trig equations

Thanks for writing to Ask Dr. Math, Veronica.

1. Use the identity tan^2(x) + 1 = sec^2(x), and convert this into
   a quadratic equation in sec(x). Solve it for sec(x). Then finish.

2. There is a general technique for equations of the form

   a*sin(x) + b*cos(x) = c.

Divide by sqrt(a^2+b^2), and figure out the angle A such that

   cos(A) = a/sqrt(a^2+b^2),
   sin(A) = b/sqrt(a^2+b^2).

Then the equation will take the form

   cos(A)*sin(x) + sin(A)*cos(x) = c/sqrt(a^2+b^2),
   sin(x+A) = c/sqrt(a^2+b^2).

(Why?) That will allow you to figure out x + A, and thus x.

In your case, a = 1 and b = -sqrt(3), so sqrt(a^2+b^2) = 2.  
Then cos(A) = 1/2, sin(A) = -sqrt(3)/2, so A = -60 degrees, or 
-Pi/3 radians. (Why?) You finish.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/