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Solving for Angle Beta (Trigonometry Problem - Type IV)

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In the attached problem I need to solve for the unknown angle Beta. The book calls it a type IV problem and gives the procedure on how to solve for a type IV problem. I have tried to solve this problem but with no luck.
I have attached the problem, the procedures to solve for the type IV and the formulas and pictue of a type I figure. I would like to be able to solve this problem the way the book explains.

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Solution Summary

The solution solves for the angle beta in the given problem.

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Please see attached file. I have used Pythagoras theorem and Cosine Rule.

If the common edge BC of two right angled triangles is considered unity, then
In ...

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