Theorem 4

General solution of acosq + bsinq = c

Method I

-----(i)

Put a = r cos a, b = r sin a -----(ii)

(i) can be written as

From (ii) and (iii), we obtain the auxiliary angle a.

Now c and r are known, and from (iv) the general solution of q is determined.

Method II

a cos q + b sin q = c -----(i)

(i) can be written as

or a (1- t²) + 2bt - c (1+t²) = 0

This is a quadratic in 't' and can be solved.

Examples:

Find the general solution of the following equations:

4) cos q - cos q = 1

Suggested answer:

We get,

4) cos q - sin q = 1

Note:

So the general solution is the same as for sin q.

So the general solution is the same as for cos q.

So the general solution is the same as for tan q.

i.e., if cot q = k, then q = np + a.