Trigonometry Word Problems

I'm having a hard time finding the answer to these three word problems while the answer is provided in the answer key I can't seem to reach the same answer so I would appreciate if the work was shown clearly so I can understand where I went wrong in my steps. Questions are in the screenshot provided.

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  • the answers are good but I would like to know why the a variable got replaced by the X and why a +1 was added in question 46

  • also what led you to choose tan over sine law or cosine law?

  • a is an auxiliary variable that helps you to solve the problem. In Problem 46, a is the distance from the center of the hill.

  • Using tan suits this problem very well. If you try to use sine or cosine rules, you are going to need distances or angles that are not given, and your solution may become much more complicated. I suggest you to try to solve it using sine or cosine rule, and you will see that your solutions is going to be messy. If you like to get better at choosing a good method/rule for solving problems, you should try to see solutions for various related problems. Then you will start to see the patterns.

  • in #46 at first it was tan37 - tan 32/ tan 32 then it was flipped to tan32 / tan 37 - tan 32 can you explain why, please?

  • in #46 at first, it was tan37 - tan 32/ tan 32 then it was flipped to tan32 / tan 37 - tan 32 can you explain why, please? -

  • I added the details at the end of the file. See the second attachment.

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The answer is accepted.