WAEC 2018 · Paper 2 · Q10

(a)

Copy and complete the table of values for $y = 2\cos x - \sin x$ , $0^\circ \le x \le 300^\circ$ .

$x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$	$270^\circ$	$300^\circ$
$y$	$2.00$		$0.13$		$-1.87$		$-2.00$		$-0.13$

Model answer

$x$	0°	30°	60°	90°	120°	150°	180°	210°	240°	270°	300°
$y$	2.00	1.23	0.13	−1.00	−1.87	−2.23	−2.00	−1.23	−0.13	1.00	1.87

For example, at $x = 30^\circ$ : $y = 2(0.866) - 0.5 = 1.23$ (2 d.p.).

(b)
Using scales of 2 cm to $30^\circ$ on the $x$ -axis and 2 cm to 1 unit on the $y$ -axis, draw the graph of $y = 2\cos x - \sin x$ for $0^\circ \le x \le 300^\circ$ .
Model answer
Plot every point from the table, then join them with one smooth curve (not straight lines between points). Scale: 2 cm to $30^\circ$ , 2 cm to 1 unit.

For (c): (i) draw $y = 1$ : it meets the curve at $x \approx 36.9^\circ$ and $270.0^\circ$ . (ii) $\tan x = 2$ means $\sin x = 2\cos x$ , i.e. $2\cos x - \sin x = 0$ : read where the curve crosses the $x$ -axis, $x \approx 63.4^\circ$ and $243.4^\circ$ .
(c)(i)
Use the graph to find the values of $x$ for which $2\cos x - \sin x = 1$ .
Separate values with commas, e.g. 3, −2
(c)(ii)
Use the graph to find the values of $x$ for which $\tan x = 2$ .
Separate values with commas, e.g. 3, −2

x in degrees. The line y = 1 gives (c)(i); the x-axis gives (c)(ii).

Worked solution (try it first)

In degree mode, to 2 decimal places as in the table.
For example, $x = 30^\circ$ : $2(0.866) - 0.5 = 1.23$ .
$x = 150^\circ$ : $2(-0.866) - 0.5 = -2.23$ .
The full row is $2.00, 1.23, 0.13, -1.00, -1.87, -2.23, -2.00, -1.23, -0.13, 1.00, 1.87$ .

Draw the line $y = 1$ and read down from where it crosses the curve: $x \approx 37^\circ$ and $x = 270^\circ$ .

Rearrange so it matches the graph: $\tan x = 2$ means $\frac{\sin x}{\cos x} = 2$ , so $\sin x = 2\cos x$ , which is $2\cos x - \sin x = 0$ .
So read where the curve crosses the $x$ -axis: $x \approx 63^\circ$ and $x \approx 243^\circ$ .