a) Copy and complete the following table of values for y = 2 cos x – sin x ,\(0^o \leq x \leq 300^o\) \[\begin{array}{c|c} x & 0^o & 30^o & 60^o & 90^o & 12...
Question 1 Report
a) Copy and complete the following table of values for y = 2 cos x – sin x ,\(0^o \leq x \leq 300^o\)
\[\begin{array}{c|c} x & 0^o & 30^o & 60^o & 90^o & 120^o & 150^o & 180^o & 210^o & 240^o & 270^o& 300^o \\ \hline Y & 2.00 & & 0.13 & & -1.87 & & -2.00 & & -0.13 & & \end{array}\]
(b) Using scales of 2 cm to 30\(^o\) on the x-axis and 2cm to 1 unit on the y-axis, draw the graph of y = 2 cos x – sin x for \(0^o \leq x \leq 300^o\)
(c) Use the graph to find the value(s) of x for which:
(i) 2 cos x – sin x = 1;
(ii) tan x = 2.
Answer Details
(a) Completed table for \(y=2\cos x-\sin x\):
| \(x\) | \(0^\circ\) | \(30^\circ\) | \(60^\circ\) | \(90^\circ\) | \(120^\circ\) | \(150^\circ\) | \(180^\circ\) | \(210^\circ\) | \(240^\circ\) | \(270^\circ\) | \(300^\circ\) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(y\) | 2.00 | 1.23 | 0.13 | -1.00 | -1.87 | -2.23 | -2.00 | -1.23 | -0.13 | 1.00 | 1.87 |
For example, at \(x=30^\circ\):
\[y=2\cos30^\circ-\sin30^\circ=2(0.866)-0.5=1.232\ldots\approx1.23.\]
(b) Graph of \(y=2\cos x-\sin x\)
(c)(i) To solve \(2\cos x-\sin x=1\), draw or imagine the horizontal line \(y=1\). The required values are the \(x\)-coordinates where this line meets the curve:
\[x\approx36^\circ\quad\text{or}\quad x\approx270^\circ.\]
(c)(ii) Since
\[\tan x=2 \quad\Rightarrow\quad \frac{\sin x}{\cos x}=2 \quad\Rightarrow\quad \sin x=2\cos x,\]
this is equivalent to
\[2\cos x-\sin x=0.\]
Therefore, read the \(x\)-coordinates where the curve crosses the \(x\)-axis:
\[x\approx63^\circ\quad\text{or}\quad x\approx243^\circ.\]
Examination reminder: For an equation involving \(2\cos x-\sin x\), use the graph’s \(y\)-value. In particular, \(\tan x=2\) rearranges to \(2\cos x-\sin x=0\), so look for the \(x\)-intercepts of the curve.
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