WAEC 2009 · Paper 2 · Q12

(a)

Copy and complete the table of values for $y = \sin x + 2\cos x$ , correct to one decimal place.

$x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$
$y$		$2.2$				$-1.2$	$-2.0$		$-1.9$

Model answer

$x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$
$y$	$2.0$	$2.2$	$1.9$	$1.0$	$-0.1$	$-1.2$	$-2.0$	$-2.2$	$-1.9$

(b)
Using a scale of 2 cm to $30^\circ$ on the $x$ -axis and 2 cm to 0.5 units on the $y$ -axis, draw the graph of $y = \sin x + 2\cos x$ for $0^\circ \le x \le 240^\circ$ .
Model answer
Plot the nine points from the table and join them with one smooth curve. Scale: 2 cm to $30^\circ$ , 2 cm to 0.5 units. The curve falls from 2 at $0^\circ$ (after a small rise to about 2.2 near $27^\circ$ ), crosses the $x$ -axis near $117^\circ$ and reaches about $-2.2$ near $207^\circ$ .

For (c) and (d): the curve crosses $y = 0$ at $x \approx 117^\circ$ ; the line $y = 2.1$ meets it at $x \approx 6^\circ$ and $47^\circ$ ; at $x = 171^\circ$ , $y \approx -1.8$ .
(c)(i)
Use your graph to solve the equation $\sin x + 2\cos x = 0$ .
(c)(ii)
Use your graph to solve the equation $\sin x = 2.1 - 2\cos x$ .
Separate values with commas, e.g. 3, −2
(d)
From the graph, find $y$ when $x = 171^\circ$ .

Try it on a graph

x in degrees. The x-axis gives (c)(i); the line y = 2.1 gives (c)(ii).

Worked solution (try it first)

(a)

(b)

(c)(i)

$\sin x + 2\cos x = 0$ where the curve crosses the $x$ -axis: $x \approx 117^\circ$ (exactly, $\tan x = -2$ gives $116.6^\circ$ ).

(ii)

Rearrange: $\sin x + 2\cos x = 2.1$ .
Draw the line $y = 2.1$ and read down where it meets the curve: $x \approx 6^\circ$ and $x \approx 47^\circ$ .

(d)