inverse trig

**talor** · August 9th 2008, 09:23 AM

(theta) = cos ^-1(x) double arrow x=cos(theta) Is this a true or false statement and if so how can I verify it?

**o_O** · August 9th 2008, 09:54 AM

$\theta = \cos^{-1} x \: \: \Rightarrow \: \: \cos \theta = x$

But not the other way around: $\cos \theta = x \: \: \not{\Rightarrow} \: \: \theta = \cos^{-1} x$ strictly speaking in the sense that the inverse cosine function has a range of $[0, \pi]$

For example, $\theta = \frac{5\pi}{4} \: , \: x = -\frac{\sqrt{2}}{2}$

$\cos \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

But: $\cos^{-1} \left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$

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