Science:Math Exam Resources/Courses/MATH100/December 2012/Question 02 (c)
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Question 02 (c) 

ShortAnswer Questions. Questions 14 are shortanswer questions. Put your answers in the boxes provided. Simplify your answers as much as possible, and show your work. Each question is worth 3 marks, but not all questions are of equal difficulty.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

This is a composition of functions. What rule helps differentiate compositions of functions? 
Hint 2 

Use the chain rule, namely

Hint 3 

A reminder that

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Since
and
The chain rule gives us

Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. For an alternative solution, taking the sin of both sides yields
Taking implicit derivatives of both sides yields
Solving for the derivative yields
Next, we want to express the right hand side as a function of x. Since and is opposite over hypotenuse, we can draw the following triangle, interpreting y as an angle: where the last side (the base of the triangle) is computed using the Pythagorean theorem. Thus we see that
and plugging into the derivative yields
completing the problem. 