(a) Using a scale of 2 cm to 30° on the x- axis, 2 cm to 0.2 units on the y- axis, on the same graph sheet, draw the graphs of \(y = \sin 2x\) and \(y = \co...

Answer Details

(a) Prepare a table of values for \(y = \sin 2x\) and \(y = \cos x\) at intervals of \(30^\circ\) for \(0^\circ \le x \le 210^\circ\). For example, at \(x = 30^\circ\): \(\sin 2x = \sin 60^\circ = 0.87\) and \(\cos x = \cos 30^\circ = 0.87\); at \(x = 120^\circ\): \(\sin 2x = \sin 240^\circ = -0.87\) and \(\cos x = \cos 120^\circ = -0.50\). The complete table is:

Plotting these points, using a scale of 2 cm to \(30^\circ\) on the \(x\)-axis and 2 cm to 0.2 units on the \(y\)-axis, gives the two smooth curves below:

(b)(i) The truth set of \(\sin 2x = 0\) is read where the curve \(y = \sin 2x\) crosses the \(x\)-axis (i.e. where \(y = 0\)). From the graph these crossings occur at \(x = 0^\circ,\ 90^\circ\) and \(180^\circ\):

(b)(ii) The truth set of \(\sin 2x - \cos x = 0\), i.e. \(\sin 2x = \cos x\), is read where the two curves intersect. From the graph the curves cross at \(x = 30^\circ,\ 90^\circ\) and \(150^\circ\).

This agrees with the algebra: \(\sin 2x = \cos x \Rightarrow 2\sin x\cos x = \cos x \Rightarrow \cos x\,(2\sin x - 1) = 0\), so \(\cos x = 0 \Rightarrow x = 90^\circ\), or \(\sin x = \tfrac{1}{2} \Rightarrow x = 30^\circ,\ 150^\circ\). Hence the truth set is: