Square of equations - Why some solutions rejected

Dear all,

I'm solving 2 sums

a) 1 - cos x = (sqrt 3) sin x

By squaring both sides, and using the sin^2x + cos^2x = 1 relation,

I get 2 cos^2x - cos x - 1=0

Solving, I get x = 120, 240 (for 2nd, 3rd quadrant) and 0, 360 (for 1st, 4th quadrant)

May I know why 240 is rejected?

b) 2 + sin x = 4 cos x

Similarly, I'm solving down to 17 sin^2x + 4 sinx - 12 = 0

I get sin x = 0.7307 or -0.966

Hence, x - 46.9 or 133.1 ; 285 or 255

Why is 133.1 and 255 rejected?

Thanks

Re: Square of equations - Why some solutions rejected

The idea is that implies for all x, but not vice versa. The correct equivalence is

.

Thus, if you solve and find that, say,

,

it does not follow that

.

At most you can guarantee that

.

Indeed, in your first example

.

Re: Square of equations - Why some solutions rejected

Hello, Rodimus79!

;808717]Dear all,

I'm solving 2 sums

Quote:

By squaring both sides, and using the sin^2x + cos^2x = 1 relation,

I get 2 cos^2x - cos x - 1=0

Solving, I get x = 120^{o}, 240^{o} (for 2nd, 3rd quadrant) and 0^{o}, 360^{o} (for 1st, 4th quadrant)

May I know why 240^{o} is rejected?

When squaring an equation, we __must__ *check our answers*.

Some answers may not check out.

. .

. .

. .

. .

Re: Square of equations - Why some solutions rejected

Hello again!

Do you know WHY squaring an equation can introduce "extra" solutions?

Consider the equation: .

The answer is, of course: .

Suppose we square the equation before solving.

Then we have: .

This is a quadratic equation; it will have two solutions.

. . .

We find that satisfies the equation,

. . but does not.

Therefore, is an *extraneous* solution.