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Find the Exact Value (No Calculator)

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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?

Apr 17, 2019
edited by MemeLord  Apr 17, 2019

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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

Apr 18, 2019
#3
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$$\text{Consider the range of }\tan^{-1} x.\\ \tan^{-1} x \in \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]\\ \text{That means we need to find a value } v\text{, which is between}-\dfrac{\pi}{2}\text{ and }\dfrac{\pi}{2}\text{, such that}\tan v = \tan\left(\dfrac{4\pi}{5}\right)\\$$

$$\text{Solving the equation, }\\ v = \dfrac{4\pi}{5} + n\pi, \; n\in \mathbb Z$$

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 $$\tan^{-1}\left(\tan\left(\dfrac{4\pi}{5}\right)\right) = \dfrac{-\pi}{5}$$

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Apr 18, 2019