Science:Math Exam Resources/Courses/MATH102/December 2012/Question A 02/Solution 1

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).