Question

from the pythagoren identites:
cos^2x=1-sin^2x => cos^2x - 1 = -sin^2x
Is it possible then to conclude that +-sqrt(cos^2-1)=-sin^2x? .... i see a possible problem here with the square root being negative

Answer 1

first of all, your conclusion has something wrong at +-sqr (cos^2 -1 = +- sinx . not sin^2.
from your equation, -sin^2x has exponent to the2 of sinx only, not for -. so, if you count like that, you take sqr (sin^2 only) to get +- sinx and then combine with original- you still have +-sinx =+-sqr(cos^2-1), I mean the sqr calculate doesn't count the -. if it has, in complex number only. and you still have trig number on complex form

Answer 2

hym, this is certainly confusing. So does that mean to say that \[- x^2 = y^2 \implies +- x = +-\sqrt(y^2)\]

Answer 3

do you know about complex number?

Answer 4

that part help you take sqr of a negative number. and every number can be expressed under that form. it has 2 part: 1st part is real and second part is unreal

Answer 5

yeah

Answer 6

when second part =0, that number become real number. when the 1st part =0, it is unreal number

Answer 7

i dont see how that applies?

Answer 8

thank you for your help :)

Answer 9

since trigs is used to count the angle . the value of sin or cos depend on the angle.right? you use the trigs to solve the problem, you give out just a circled problem like that , how can you conclude some thing to apply to some where?

Answer 10

If you have a particular problem, which need that conclusion to solve, post it up. If I don't know, i will ask my professor. anyway, your question is interesting. wait. next week, i will have the 100% right answer for you after i get the confirmation from my pro. don't close the question. is it ok?

Answer 11

ok i wont close it. It seems as though my steps were valid in arriving at cos^2-1 = -sin^2, the main question is then can we solve for sin and what would the solution look like :)

Answer 12

Hi, here is my professor's answer, hope you can find out what you need, I copy the whole thing for you.
The question is ambiguous, unless there is an agreed-upon definition for square root. When dealing with number which may not be positive, "square root" sometimes means "any number whose square is ..." (so there would be two candidates unless the number is 0) or possibly "all numbers whose square is ..." (so the symbol would actually represent a set of two numbers). The problem with this definition is that an equation like sqrt(4) + sqrt(4) = 0 is hard to say that it's false, because both symbols represent {2,-2} and so 0 is one of the sums.

With positive numbers, you can avoid this and make "square root" a function from the nonnegative numbers to the real numbers. You can do that in infinitely many ways, by choosing one of the two solutions of x^2 = number for each positive number. If you always choose the positive one, you get the usual meaning for "square root" -- the positive solution of x^2 = number.

But that option is not available for numbers which might not be positive. If you try to make a choice anyway, you have another problem: the resulting function is not multiplicative. No matter how you define sqrt(-1) for example, you can't have

sqrt(-1) * sqrt(-1) = sqrt((-1)*(-1)) = sqrt(1)

because this would mean you would have to define sqrt(1) = -1, and then

sqrt(1) * sqrt(-1) = (-1) sqrt(-1) but also
sqrt(1) * sqrt(-1) = sqrt(1 * (1-)) = sqrt(-1)

The "square root of a negative number", as in a number whose square is negative, cannot be a *real* number. So what's left?

The moral here is to not be sloppy with functions: you need to know their domain and codomain and actual definition as a function (pick one of the two solutions to x^2 = number). If the equation has no solutions, then "number" cannot be in the domain.

The equation sqrt(cos^2(x) - 1) = sqrt(-sin^2(x)) is true in the sense that

"the solutions of u^2 = cos^2(x) - 1 are the same as the solutions of u^2 = -sin^2(x)"

whether or not we consider the equation to have any solutions, but false if "sqrt" is defined as a function from nonnegative real numbers to nonnegative real numbers (the positive square root)

Answer 13

im thinking we might just have to get rid of the negative by conventional means, ide like to understand the properties of the statement -x^2=t^2 with out getting rid of the negative but im beginning to think its a childish question, because its simple to divide out the negative to x^2=-t^2

Answer 14

AH! i didnt even realize what i had gotten myself into! lol. Now i see how this is related to complex numbers! :) Thank you for asking your professor about this, i have certainly learned alot from this thread

Answer 15

you're welcome. we have to make clear what we are not sure. have a good day

Answer 16

This makes me wonder if Euler took a similar route in finding the significance of i*sinx