Question

Quick silly math question

Answer 1

When you square root something like\[\large \sqrt{\sin^2 \theta}\] will it be \[\large \pm \sin \theta\]or\[\large |\sin \theta|\]

Answer 2

absolute value of sin because the result of a positive square root must be positive.

Answer 3

Those are the same thing,\[\large\rm \pm \sin \theta=|\sin \theta|\]

But you do bring up a good point,
why don't we have an absolute value when we trig substitutions in integration?
Example:\[\large\rm \int\limits\frac{3x+2}{\sqrt{x^2+1}}dx\]we would make the substitution,\[\large\rm x=\tan \theta\]Which would turn the denominator into \(\large\rm \sqrt{\sec^2\theta}\) which for some weird reason gives us \(\large\rm \sec\theta\) not \(\large\rm |\sec\theta|\).

Answer 4

I don't actually know the answer to the question I posed,
someone explained it well on a stackexchange if I can find it.

Answer 5

You can also think about the graph of the generalised
\[y = x^2 \]

All ordinates will be positive so taking the sqrt of positive ordinates gives a positive result

Answer 6

@zepdrix no, not quite. this is a common misconception. You are right ONLY if x follows the following conditions...

|sinx| = sin x when sin x is positive or 0
|sinx| = -sinx when sinx is negative

range of sqrt(f(x)) is positive or zero.

sqrt(sin^2 x) intrinsically has sin^2 x being positive to start with.
writing sqrt(sin^2 x) = sin x is correct when sin x is positive and writing
sqrt(sin^2 x) = - sin x is correct only when sin x is negative.

The absolute value groups these two conditions together.

However, writing\[\sqrt{\sin^2 x} = \pm \sin x\] is not correct because it does not apply the restrictions on x to make the sqrt function positive.

Answer 7

Graph

https://www.wolframalpha.com/input/?i=plot+sqrt(sin%5E2(x))

if you have \[y = \pm \sin x \] you will not have a function.

Answer 8

Oh sorry I guess I thought that was obvious/implied when I wrote \(\large\rm |\sin\theta|=\pm\sin\theta\) that I meant it with the restrictions.

Answer 9

yeah my high school senior mathematics teacher was very pedantic, as was my first year calc lecturer in defintion, cos many students came with a very thin understanding the definition of absolute value along the lines of 'it's always positive, or it gives positive and negative of the number'.

Answer 10

√16 means by definition the positive square root of 16. That is 4 and only 4.

In an equation such as x² = 16, note that you are asked to find numbers which when squared equal 16. This is a different question from x = √16, x = ?

x² = 16

x² - 16 = 0

( x + 4) * (x -4) = 0

x + 4 = 0 and x - 4 = 0

x = -4, x = 4

Therefore there are two solutions, both the positive and negative square roots of 16.