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 April 18th, 2018, 05:09 AM #1 Newbie   Joined: Apr 2018 Posts: 2 Thanks: 0 Pove Hi how can prove cos-1(-x)=-cos-1x Sent from my iPhone using Tapatalk
 April 18th, 2018, 07:14 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,183 Thanks: 870 If you cannot use Latex, at least use "^" to indicate the inverse function. You want to prove that cos^(-1)(x)= -cos^(-1)(-x) ($\displaystyle cos^{-1}(x)= -cos^{-1}(-x)$. Your basic problem is that you can't prove that, it isn't true! For example, if $\displaystyle x= \frac{\sqrt{2}}{2}$ then $\displaystyle cos^{-1}(x)= \frac{\pi}{4}$ radians (45 degrees). But the definition of the function $\displaystyle cos^{-1}(x)$ restricts the range to between 0 and $\displaystyle \pi$ (180 degrees). $\displaystyle cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)= \frac{3\pi}{4}$ (135 degrees), NOT $\displaystyle -\frac{\pi}{4}$ radians or -45 degrees. If you were to change "$\displaystyle cos^{-1}$" to "$\displaystyle sin^{-1}$" then it would be true.
April 18th, 2018, 09:23 AM   #3
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Quote:
 Originally Posted by zryanomer45 Hi how can prove cos-1(-x)=-cos-1x Sent from my iPhone using Tapatalk
A correct equation that may be proven could be ...

$\cos^{-1}(-x) = \pi - \cos^{-1}(x)$

April 18th, 2018, 09:45 AM   #4
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Quote:
 Originally Posted by skeeter A correct equation that may be proven could be ... $\cos^{-1}(-x) = \pi - \cos^{-1}(x)$

Ok how can l prove it

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 April 18th, 2018, 10:05 AM #5 Senior Member     Joined: Sep 2015 From: USA Posts: 1,974 Thanks: 1025 $\cos^{-1}(-x) = -\cos^{-1}(x)$ $-x = cos(-\cos^{-1}(x))$ the cosine is an even function so $\cos(-x)=\cos(x)$ thus $-x = \cos(\cos^{-1}(x)) = x$ and thus as Country Boy keenly noted your statement isn't true.
 April 18th, 2018, 10:10 AM #6 Math Team   Joined: Jul 2011 From: Texas Posts: 2,755 Thanks: 1405 evaluate the cosine of each side ... $\cos[\cos^{-1}(-x)] = -x$ $\cos[\pi - \cos^{-1}(x)] = \cos(\pi)\cos[\cos^{-1}(x)] + \sin(\pi)\sin[\cos^{-1}(x)] = (-1)(x) + 0 = -x$

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