Science:Math Exam Resources/Courses/MATH110/December 2010/Question 10
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Question 10 

(originally for bonus marks) Prove that the curve has an infinte number of horizontal tangent lines. 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

A curve has horizontal tangent lines when its derivative is equal to zero. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. As was noted in the hint, the curve has horizontal tangent lines at values where its derivative is equal to zero. Thus we first take the derivative of the curve, using the quotient rule:
We set this equal to zero and simplify to get:
So we are trying to find xvalues that satisfy this final equation; if such an xvalue exists, has a horizontal tangent at that point. From here, there is no specific method or surefire trick to find the answer  the only thing required is an understanding of the behavior of and , especially considering their values as points on the unit circle. The most useful observation here is that on the unit circle, when is zero, is and vice versa. This is one step towards a solution, because if we can find a place where and , we will have found a combination of and that will equal 1, which is what we want. In fact, when we know that and . On the right hand side of the equation above this gives 1, just like we want. Thus is one solution to our equation; our curve will have a horizontal tangent line here. Because and are periodic, we know that this solution will work anytime we are at the same angle on the unit circle, i.e. the angles and so on. So there are an infinite number of solutions, and so the curve has an infinite number of horizontal tangent lines. 