Science:Math Exam Resources/Courses/MATH102/December 2012/Question A 02

MATH102 December 2012

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Question A 02
To find a point (a,cos(a)) on the graph of y = cos(x) whose tangent line goes through the origin, which of the following equations must you solve? (a) $\cot(a)=a$ (b) $\cot(a)=-a$ (c) $\tan(a)=a$ (d) $\tan(a)=-a$

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
Try sketching this scenario: a tangent line to $\cos(x)$ that passes through the origin. How could you compute the slope of that line? (There will be two methods)

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. As stated in the hint, we will begin by sketching a picture of $\cos x$ and a few tangent lines. The scenario described in the question is shown by the red line line. Redrawing and relabeling the sketch looks like this: (Solution writer's note: The sketches above are nice and neat for clarity, but in real life, i.e. when you're actually taking an exam, messy is okay. My original sketch to solve the problem looked like the thumbnail on the right. . The point of drawing a sketch is often just to understand what the problem is asking.) Now there are two ways to compute the slope of the tangent line above. The first is our traditional slope formula: $m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {\cos(a)-0}{a-0}}={\frac {\cos(a)}{a}}$ The second is through the derivative, as this is the tangent line of $y=\cos x$ at x = a: $\displaystyle y'=-\sin x$ $\displaystyle m=-\sin(a)$ As these two methods are giving the slope of the same line, they must be equal: ${\frac {\cos(a)}{a}}=-\sin(a)$ Cross-multiplying gives: ${\frac {\cos(a)}{\sin(a)}}=-a$ Or $\cot(a)=-a$, which is answer (b).

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