Is there a substitution method to solve pure trigonometric equations?

[Al-Layth]

I remember browsing about techniques of solving trigonometric equations and seeing somebody use a substitution of the form
[math]t=\sin(x)[/math]
or i think that was it, but i am not sure. Essentially they converted the trigonometric equation into a polynomial equation, solved the polynomial and then used that to help solve the trig equation. does anybody know what im talking about?

 

[skeeter]

Something like this simple example?

[imath]\sin^2{x} + 1 = 3\sin{x}[/imath]

Let [imath]t = \sin{x}[/imath] and solve the resulting quadratic?

 

[khansaheb]

I remember browsing about techniques of solving trigonometric equations and seeing somebody use a substitution of the form
[math]t=\sin(x)[/math]
or i think that was it, but i am not sure. Essentially they converted the trigonometric equation into a polynomial equation, solved the polynomial and then used that to help solve the trig equation. does anybody know what im talking about?

This method - converting the trigonometric equation into a polynomial equation - is a "major" trick used in solving integrals with trigonometric functions.

 

[skeeter]

This method - converting the trigonometric equation into a polynomial equation - is a "major" trick used in solving integrals with trigonometric functions.

actually, the other way around (trig substitutions for algebraic integrands) is more common

 

[Al-Layth]

Something like this simple example?

[imath]\sin^2{x} + 1 = 3\sin{x}[/imath]

Let [imath]t = \sin{x}[/imath] and solve the resulting quadratic?

there was a definition in t for both sin and cos

i forgot if it was
t= sin^2(x) or t=sin(x)

or something else.
But then the pythagorean identity was used to determine cos(x) in terms of t.

 

[khansaheb]

actually, the other way around (trig substitutions for algebraic integrands) is more common

Substituting algebraic polynomials for trig integrals is less common but not non-existent.

Anyway I was just providing examples......

 

[nasi112]

there was a definition in t for both sin and cos

i forgot if it was
t= sin^2(x) or t=sin(x)

or something else.
But then the pythagorean identity was used to determine cos(x) in terms of t.

In fact, you have the freedom to assign \(\displaystyle t\) to any trigonometric function that will help you reduce or solve a big problem. Usually, assigning \(\displaystyle t\) to trigonometric functions allow you to simplify things faster.