Obliterating Iterating (Fazed by Phase)

Date: 10/10/2003 at 11:46:16
From: Howard 
Subject: Trig Equation

What is the solution for angle [theta] where:

  a*sin[theta] + b*cos[theta] = c

with a, b and c being constants?

I can find a solution by iteration, but intuitively it seems that 
there must be a better way.

This is a generalisation of a real-world problem to find the necessary 
final inclination of a line attached to a circular arc reaching a 
defined lateral and vertical displacement.

Date: 10/10/2003 at 12:17:10
From: Doctor George
Subject: Re: Trig Equation

Hi Howard,

Thanks for writing to Doctor Math.

Although your question wasn't asked in the context of phase calculations, 
that is a common place for this question to come up.

First divide though by sqrt(a^2 + b^2) to get

         a                            b
  --------------- sin[theta] + --------------- cos[theta]
  sqrt(a^2 + b^2)              sqrt(a^2 + b^2)
                                                           c  
                                                  = ---------------
                                                    sqrt(a^2 + b^2)

Now define alpha such that

                      a                                    b
  cos[alpha] = ---------------   and   sin[alpha] = ---------------
               sqrt(a^2 + b^2)                      sqrt(a^2 + b^2)

Now we have

                                                       c
  cos[alpha]sin[theta] + sin[alpha]cos[theta] = ---------------
                                                sqrt(a^2 + b^2)

Can you finish it from there?  Write again if you need more help.

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

Date: 10/13/2003 at 13:32:04
From: Howard Moles
Subject: Thank you (Trig Equation)

Thanks Dr George!

I now recognise the solution, but it has been almost 40 
years since I first saw it.

Howard