(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 \(\frac{2}{3} \cos x + \sin x = \frac{5}{6}\).
(a)
| x | 0° | 30° | 60° | 90° | 120° | 150° | 180° | 210° |
| y | 2.00 | 3.23 | 3.60 | 3.00 | 1.60 | -0.23 | -2.00 | -3.23 |
(b) Scale : 2 cm to 60° on the x- axis
2 cm to 2 units on y- axis.
(c)(i) Maximum value of y is 3.60.
(ii) Graph : \(\frac{2}{3} \cos x + \sin x = \frac{5}{6}\)
Equation : \(2 \cos x + 3 \sin x = y\)
i.e. \(\frac{2}{3} \cos x + \sin x = \frac{y}{3} .... (2)\)
(2) - (1) : \(\frac{y}{3} - \frac{5}{6} = 0\)
\(\frac{y}{3} = \frac{5}{6} \implies y = 2.50\)
We draw the line y = 2.5 and find the x- values at the points of intersection of graph with y = 2.5.
x = 12° and 102°.
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