WAEC 2016 · Paper 2 · Q12

1. (a)

   Copy and complete the table of values, correct to one decimal place, for the relation $y = 3\sin x + 2\cos x$ for $0^\circ \le x \le 360^\circ$.

   $x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$	$270^\circ$	$300^\circ$	$330^\circ$	$360^\circ$
   $y$	$2.0$			$3.0$	$1.6$		$-2.0$		$-3.6$	$-3.0$			$2.0$

   Model answer

   $x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$	$210^\circ$	$240^\circ$	$270^\circ$	$300^\circ$	$330^\circ$	$360^\circ$
   $y$	$2.0$	$3.2$	$3.6$	$3.0$	$1.6$	$-0.2$	$-2.0$	$-3.2$	$-3.6$	$-3.0$	$-1.6$	$0.2$	$2.0$

   Work in degree mode, to one decimal place: for example $x = 30^\circ$: $3(0.5) + 2(0.866) = 3.2$, and $x = 150^\circ$: $1.5 - 1.73 = -0.2$.

2. (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 the relation.

   Model answer

   Plot every point from the table, then join them with one smooth curve (not straight lines between points). The curve peaks at about $3.6$ near $56^\circ$ and dips to about $-3.6$ near $236^\circ$.

   For (c)(i): the roots of $3\sin x + 2\cos x = 0$ are where it crosses the $x$-axis, $x \approx 146.3^\circ$ and $326.3^\circ$. For (c)(ii): $2 + 2\cos x + 3\sin x = 0$ means $y = -2$, which meets the curve at $x = 180.0^\circ$ and $292.6^\circ$.

3. (c)(i)

   Use the graph to solve $3\sin x + 2\cos x = 0$.

   Check

   Separate values with commas, e.g. 3, −2
4. (c)(ii)

   Use the graph to solve $2 + 2\cos x + 3\sin x = 0$.

   Check

   Separate values with commas, e.g. 3, −2

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