Solving trigonometric equations by introducinganAuxiliaJ)' argument. Equation of the form of

`a cos theta + b sin theta=c`

To solve equation, we convert the equation to the form `cos theta = cos alpha ` or `sin theta = sin alpha ` etc.


Forth is let us suppose that `(tt ( (a= r cos phi),(b= r sin phi) )}` `=> ` ` {tt( (r = sqrt (a^2 +b^2)),(text(and ) tan phi =b/a))`

Substituting these values in the equation `a cos theta + b sin theta =c` , we have

`r cos phi cos theta + r sin phi sin theta =c`

`=> r cos (theta - phi)=c`

`=> cos (theta - phi) = c/r = c/ (sqrt (a^2 +b^2)) = cos beta ` (suppose)

`=>theta - phi = 2 n pi pm beta`

`=> theta = 2n pi + phi =m beta, n in Z`

Here `phi` and `beta` are known as a, band care given.

Hence, we can solve the equation of this type by putting.

`a = r cos phi ` and `b= r sin phi` provided `| c/ (sqrt (a^2 + b^2)) | le 1` [`:.` `cos beta` lies between `-1` and `1`]

or `|c|/(sqrt (a^2+ b^2)) le 1` or `|c| le sqrt (a^2+ b^2)`

Solving equations by transforming a sum of trigonometric functions into a product.

Solving equations by transforming a product of trigonometric functions into a sum.

Solving equations by a change of variable or by substitution method:

(i) Equations of the form `P (sin x pm cos x, sin x* cos x)=0` , where `P(x,z)` is a polynominal,
can be solved by the change `cos x pm sin x =t`

`=> 1 pm 2 sin x* cos x=t^2` . Consider th e equation ; `sinx + cos x =1+ sinx *cos x`.

(ii) Many equations can be solved by introducing a new variable e.g. consider the equation
`sin^4 2x + cos^4 2x = sin 2x. cos 2x`.

Solving equations with the use of boundedness of the function.

Remember:-
`-1 le sinx le 1, -1 le cos x le 1, tan x in R, cotx in R`.

`| cosec x| ge 1, |sec x| ge 1`.

Solution of trigonometric equation of the form `f(x) = sqrt (phi (x))`.

(i) `f(x) ge 0, phi (x) ge 0` (ii) `f^2 (x) = phi (x)`