How do you solve sin squared theta plus cos theta equals sin theta plus cos squared theta?

Answer 1

For simplicity's sake, X represent theta.

This is the original problem: sin2x+ cosX = cos2X + sinX

This handy-dandy property is key for all you trig fanatics: sin2x+ cos2x = 1

With this basic property, you can figure out that

sin2 x=1-cos2x

and

cos2x= 1-sin2x

So we can change the original problem to:

1-cos2x+cosx = 1-sin2X + sinX

-cos2x + cosx =-sin2x + sinX

Basic logic tells you that one of two things are happening.

sin2x is equal to sinx AND cos2x is equal to cosx. The only two numbers that are the same squared as they are to the first power are 1 and 0. X could equal 0, which has a cosine of 1 and a sine of 0, or it could equal pi/2, which has a cosine of 0 and a sine of 1.

The other possibility whatever x (or theta) is, it's sine is equal to its cosine. This happens twice on the unit circle, once at pi/4 and once at 5pi/4.

If you're solving for all possible values for x and not just a set range on the unit circle, then the final solution is:

x=0+2pin x=pi/2+2pin x= pi/4 +2pin x=5pi/4+2pin (note that n is a variable)