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When doing inverse trigonometric equations on a calculator, only the lowest positive solution is shown. How can I see alternative solutions (specifically for a Casio FX85ES)?

Current behaviour: $$sin^{-1}{0.5} = 30$$

Desired behaviour: $$sin^{-1}{0.5} = 30 , 150 , 390 , ...$$

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The arcsin function only takes values between $-\pi/2$ and $\pi/2$ (or between -90 and 90 if you are using degrees). The other values are not values taken by arcsine. –  Arturo Magidin Jun 7 '11 at 18:45
I don't know about your calculator, but if mine told me $30=150$ then I would throw it away... –  mac Jun 7 '11 at 18:46
@mac: Sorry, bad notation there, I'll change it. –  Josh Jun 8 '11 at 12:00
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1 Answer

up vote 3 down vote accepted

I believe your energy would be better spent just taking the value given from the calculator and thinking about this simple observation:

In terms of the unit circle: Note that $\sin(\theta)$ refers to the $y$-value of the point on the unit circle corresponding to angle $\theta$ (Here, $\theta$ will be measured in degrees to match the OP's question, although I think radians are preferably). Unless the angle is $90^{\circ}$ or $270^{\circ}$, there are exactly two points on the unit circle with that $y$-value. Therefore, to find the other one, you simply look at the opposite angle on the circle: $180^{\circ} - \theta$. Therefore, your calculator gives that $\sin^{-1}(1/2) = 30^{\circ}$, and we quickly get the next value of $150^{\circ}$. To get the remaining values, we add multiples of $360^{\circ}$ to $30^{\circ}$ and $150^{\circ}$.

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Although this doesn't tell me how to solve the problem using a calculator it does explain how to easily solve the problem mentally; I agree that this is probably a good idea. Thanks. –  Josh Jun 11 '11 at 22:30
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