WAEC 2008 · Paper 2 · Q12

1. (a)

   Copy and complete the table of values for $y = 3\sin x + 2\cos x$ for $0^\circ \le x \le 360^\circ$.

   $x$	$0^\circ$	$60^\circ$	$120^\circ$	$180^\circ$	$240^\circ$	$300^\circ$	$360^\circ$
   $y$	$2.00$						$2.00$

   Model answer

   $x$	$0^\circ$	$60^\circ$	$120^\circ$	$180^\circ$	$240^\circ$	$300^\circ$	$360^\circ$
   $y$	$2.00$	$3.60$	$1.60$	$-2.00$	$-3.60$	$-1.60$	$2.00$

   The values from $180^\circ$ to $360^\circ$ are the negatives of those from $0^\circ$ to $180^\circ$, since $\sin$ and $\cos$ both change sign after $180^\circ$.

2. (b)

   Using a scale of 2 cm to $60^\circ$ on the $x$-axis and 2 cm to 1 unit on the $y$-axis, draw the graph of $y = 3\sin x + 2\cos x$ for $0^\circ \le x \le 360^\circ$.

   Model answer

   Plot $(0^\circ, 2.00)$, $(60^\circ, 3.60)$, $(120^\circ, 1.60)$, $(180^\circ, -2.00)$, $(240^\circ, -3.60)$, $(300^\circ, -1.60)$, $(360^\circ, 2.00)$ and join them with one smooth wave. The curve peaks at about $3.6$ near $x = 56^\circ$, crosses the $x$-axis near $146^\circ$ and $326^\circ$, and is lowest, about $-3.6$, near $236^\circ$.

3. (c)

   Use your graph to solve the equation $3\sin x + 2\cos x = 1.5$.

   Check

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

   Find the range of values of $x$ for which $3\sin x + 2\cos x < -1$.

   Check

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

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