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Find the value of $\sqrt{\left(\cos\theta\right)^2}$ at $\theta=\pi$

There can be two ways in which people will do this:

  • First way: Put the value of theta directly, so $\sqrt{\left(\cos\pi\right)^2}$, answer will be $1$

  • Second way: Simplify the expression first and then put the value of $\theta$, so in this case one will get $-1$.

Why do these two ways give different answers?

Blue
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user3290550
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    They are not. When we write $\sqrt{x}$ we always mean the positive square root. So we can't write $\sqrt{x^2}=x$, this is not true in general. It should be $\sqrt{x^2}=|x|$. – Mark Oct 23 '19 at 21:06
  • Answer will be 1, you must have done mistake in second way – Zenix Oct 23 '19 at 21:07
  • @zenix, its simple to see why the answer is $-1$ in the second way, $\sqrt{\left(\cos\theta\right)^2}=\cos\theta$, now put the value of theta as $\pi$, you will get $-1$ – user3290550 Oct 23 '19 at 21:09
  • It will be $|cos\theta|$ – Zenix Oct 24 '19 at 14:45

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The existence of two correct answers is incorrect, because $\sqrt{(\cos\theta)^2}$ is the principal root of $\cos\theta,$ so it must be positive. In other words, it is equivalent to the absolute value of $\cos\theta.$ It is important not to confuse the solution to an equation like $\cos^2\theta = 1\Rightarrow \cos\theta=\pm 1$ with the principal square root as $\cos\theta$ does not have to be positive in the latter equation.

Simon Fraser
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