mat 2010 Q3

mat · Uk Standard trigonometric equations Count zeros or intersection points involving trigonometric curves
3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.
Computer Science applicants should turn to page 14. [0pt] [In this question, you may assume that the derivative of $\sin x$ is $\cos x$.] [Figure]
(i) In the diagram above $O A$ and $O C$ are of length 1 and subtend an angle $x$ at $O$. The angle $B A O$ is a right angle and the circular arc from $A$ to $C$, centred at $O$, is also drawn.
By consideration of various areas in the above diagram, show, for $0 < x < \pi / 2$, that
$$x \cos x < \sin x < x .$$
(ii) Sketch, on the axes provided on the opposite page, the graph of
$$y = \frac { \sin x } { x } , \quad 0 < x < 4 \pi$$
Justify your value that $y$ takes as $x$ becomes small. [0pt] [You do not need to determine the coordinates of the turning points.]
(iii) Drawn below is a graph of $y = \sin x$. Sketch on the same axes the line $y = c x$ where $c > 0$ is such that the equation $\sin x = c x$ has exactly 5 solutions. [Figure]
(iv) Draw the line $y = c$ on the axes on the opposite page.
(v) If $X$ is the largest of the five solutions of the equation $\sin x = c x$, explain why $\tan X = X$. [Figure]
(i) [4 marks] Let $x$ denote the angle substended by the two sides of length 1 . The area of the smaller triangle is $\frac { 1 } { 2 } \sin x$, of the sector is $\frac { 1 } { 2 } x$ and of the larger triangle is $\frac { 1 } { 2 } \tan x$. So $\sin x < x < \tan x$. Multiplying the second inequality by $\cos x > 0$ (for $0 < x < \frac { \pi } { 2 }$ ) we have $x \cos x < \sin x < x$. [0pt] 
\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science applicants should turn to page 14.\\[0pt]
[In this question, you may assume that the derivative of $\sin x$ is $\cos x$.]\\
\includegraphics[max width=\textwidth, alt={}, center]{f229588a-5602-44a5-acb0-6efec65f41af-10_387_641_712_705}\\
(i) In the diagram above $O A$ and $O C$ are of length 1 and subtend an angle $x$ at $O$. The angle $B A O$ is a right angle and the circular arc from $A$ to $C$, centred at $O$, is also drawn.

By consideration of various areas in the above diagram, show, for $0 < x < \pi / 2$, that

$$x \cos x < \sin x < x .$$

(ii) Sketch, on the axes provided on the opposite page, the graph of

$$y = \frac { \sin x } { x } , \quad 0 < x < 4 \pi$$

Justify your value that $y$ takes as $x$ becomes small.\\[0pt]
[You do not need to determine the coordinates of the turning points.]\\
(iii) Drawn below is a graph of $y = \sin x$. Sketch on the same axes the line $y = c x$ where $c > 0$ is such that the equation $\sin x = c x$ has exactly 5 solutions.\\
\includegraphics[max width=\textwidth, alt={}, center]{f229588a-5602-44a5-acb0-6efec65f41af-10_490_753_1857_648}\\
(iv) Draw the line $y = c$ on the axes on the opposite page.\\
(v) If $X$ is the largest of the five solutions of the equation $\sin x = c x$, explain why $\tan X = X$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f229588a-5602-44a5-acb0-6efec65f41af-11_801_1301_237_397}\\
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q6 Q7