+81 Find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a
calculator.
\(tan^{-1}(tan(\frac{4\pi}{5}))\). I would know how to do it, other than the fact that \(\frac{4\pi}{5}\) isn't on the unit circle. How would I solve this without a calculator?
+37157 The tan inverse and the tangent just 'cancel' each other and you are left with 4pi/5....
DO you see? First you do the tangent.....then you do the tangent inverse...
here is another example sin^-1 ( sin 30) = sin^-1 ( 1/2 ) = 30
+9674 \(\text{Consider the range of }\tan^{-1} x.\\ \tan^{-1} x \in \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]\\ \text{That means we need to find a value } v\text{, which is between}-\dfrac{\pi}{2}\text{ and }\dfrac{\pi}{2}\text{, such that}\tan v = \tan\left(\dfrac{4\pi}{5}\right)\\ \)
\(\text{Solving the equation, }\\ v = \dfrac{4\pi}{5} + n\pi, \; n\in \mathbb Z\)
Only when n = -1, v is inside the required range.
The solution we need is \(v = \dfrac{4\pi}{5} - \pi = \dfrac{-\pi}{5}\).
Therefore \(\tan^{-1}\left(\tan\left(\dfrac{4\pi}{5}\right)\right) = \dfrac{-\pi}{5}\)
