(a) Copy and complete the table of the relation \(y = 2\sin x - \cos 2x\). x 0° 30° 60° 90° 120° 150° 180° y 0.5 -1.0 Using a scale of 2 cm to 30° on the x-...
Question 1 Report
(a) Copy and complete the table of the relation \(y = 2\sin x - \cos 2x\).
| x | 0° | 30° | 60° | 90° | 120° | 150° | 180° |
| y | 0.5 | -1.0 |
Using a scale of 2 cm to 30° on the x- axis and 2 cm to 0.5 unit on the y- axis, draw the graph of \(y = 2\sin x - \cos 2x\), for \(0° \leq x \leq 180°\).
(b) Using the same axes, draw the graph of \(y = 1.25\).
(c) Use your graphs to find the : (i) values of x for which \(2\sin x - \cos 2x = 0\) ; (ii) the roots of the equation \(2\sin x - \cos 2x = 1.25\).
Answer Details
(a) For \(y=2\sin x-\cos 2x\), the completed table is:
| \(x\) | \(0^\circ\) | \(30^\circ\) | \(60^\circ\) | \(90^\circ\) | \(120^\circ\) | \(150^\circ\) | \(180^\circ\) |
|---|---|---|---|---|---|---|---|
| \(y\) | \(-1.0\) | \(0.5\) | \(2.2\) | \(3.0\) | \(2.2\) | \(0.5\) | \(-1.0\) |
For example,
\[y(60^\circ)=2\sin60^\circ-\cos120^\circ=2\left(\frac{\sqrt3}{2}\right)-\left(-\frac12\right)=2.232\ldots\approx2.2.\]
(b) The graph of \(y=1.25\) is the horizontal line shown on the same axes.
(c)
(i) The curve cuts the \(x\)-axis at approximately
\[\boxed{x=21^\circ\text{ and }159^\circ}.\]
(ii) The intersections of the curve with \(y=1.25\) give
\[\boxed{x=42^\circ\text{ and }138^\circ}.\]
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