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### 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.

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
```
Associated Topics:
College Trigonometry

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