(a) Copy and complete the table for the relation: \(y = 2\cos x + 3\sin x\) for \(0° \leq x \leq 360°\). x 0° 30° 60° 90° 120° 150° 180° 210° y 2.00 3.23 1....
Question 1 Report
(a) Copy and complete the table for the relation: \(y = 2\cos x + 3\sin x\) for \(0° \leq x \leq 360°\).
| x | 0° | 30° | 60° | 90° | 120° | 150° | 180° | 210° |
| y | 2.00 | 3.23 | 1.60 | -3.23 |
(b) Using a scale of 2 cm to 60° on the x- axis and 2 cm to one unit on the y- axis, draw the graph of \(y = 2\cos x + 3\sin x\) for \(0° \leq x \leq 360°\).
(c) From the graph, find the : (i) maximum value of y, correct to two decimal places ; (ii) solution of the equation \(\frac{2}{3}\cos x + \sin x = \frac{5}{6}\).
Answer Details
(a) For each value of \(x\), calculate \(y=2\cos x+3\sin x\), correct to two decimal places.
| \(x\) | \(0^\circ\) | \(30^\circ\) | \(60^\circ\) | \(90^\circ\) | \(120^\circ\) | \(150^\circ\) | \(180^\circ\) | \(210^\circ\) |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 2.00 | 3.23 | 3.60 | 3.00 | 1.60 | -0.23 | -2.00 | -3.23 |
For example,
\[y(60^\circ)=2\cos60^\circ+3\sin60^\circ=2(0.5)+3(0.8660)=3.60.\]
(b) The required graph is shown below. The plotted curve is smooth, with the horizontal scale \(2\text{ cm}\) to \(60^\circ\) and vertical scale \(2\text{ cm}\) to 1 unit.
(c)(i) The maximum ordinate of the curve is
\[\boxed{3.61}\]
correct to two decimal places.
(c)(ii) Since
\[\frac{2}{3}\cos x+\sin x=\frac56,\]
multiplying by 3 gives
\[2\cos x+3\sin x=2.50.\]
Thus, draw the line \(y=2.50\) on the graph and read the abscissae of its intersections with the curve:
\[\boxed{x\approx10^\circ\text{ or }102^\circ}.\]
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