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Question
How to find angles equal to inverse trigonometric functions e.g arc sine 4/5, arc cos 3/5. I also wish to know the exact procedure of calculating these values.I will also like to know how actually can we find the values of trigonometric functions e.g.sine 37, cos37 etc, without using trigonometric tables.
         I will definitely be highly obliged for this kindness

Answer
arctan 37 deg
arctan 37 deg  
Calculating trigonometric functions is, as the Wikipedia website puts it, "complicated".  There aren't really any simple ways to generate the trig functions or their inverses, although people have studied them since antiquity. The only way to calculate them is to use approximations in the form of terms in a series (or repeated fractions, etc.) that are increasingly more accurate as the number of terms increases. The functions in your calculator use algorithms based on these approximations.

If you want to crank out an approximate, but entirely useful, value yourself, your best bet is probably to use a Taylor series for arctan (=tan^-1), since this is accurate over the angles you are interested in (i.e., in the first quadrant from 0 to 90 degrees or 0 to pi/2 radians). The attached image shows the formula for the inverse tangent and numbers calculated using this formula for various numbers of terms, n, in the series. As you can see, the approximation converges pretty rapidly (I used the formula for the ratio of sides x/y = 3/4 so that  z < 1 as required for the formula; subtracting the answer form pi/2 gives the answer for 4/3 -> 37 degrees that you are interested in).

There are all sorts of ways to come up with formulas for this angle. Also, I suggest you investigate Taylor series expansions of functions, where the formula I presented comes from, since they have application in many practical areas.

Randy

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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