(a) Copy and complete the table of values for the relation \(y = 3 \sin 2x\). x \(o^o\) \(15^o\) \(30^o\) \(45^o\) \(60^o\) \(75^o\) \(90^o\) \(105^o\) \(12...
Question 1 Report
(a) Copy and complete the table of values for the relation \(y = 3 \sin 2x\).
| x | \(o^o\) | \(15^o\) | \(30^o\) | \(45^o\) | \(60^o\) | \(75^o\) | \(90^o\) | \(105^o\) | \(120^o\) | \(135^o\) | \(^o\) |
| y | 0.0 | 1.5 | -2.6 |
(b) Using a scale of 2 cm to 15° on the x-axis and 2cm to I unit on the y-axis, draw the graph of \(y = 3 \sin 2x\) for \(0° \geq x \geq 150°\).
(c) Use the graph to find the truth set of;
(i) \(3 \sin 2x + 2 = 0\);
(ii ) \(\frac{3}{2} \sin 2x = 0.25\).
Answer Details
(a) Completing the table for \(y=3\sin 2x\) (angles in degrees). For example \(x=30^\circ:\;3\sin 60^\circ=3(0.866)=2.6\).
| x | 0 | 15 | 30 | 45 | 60 | 75 | 90 | 105 | 120 | 135 | 150 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y | 0.0 | 1.5 | 2.6 | 3.0 | 2.6 | 1.5 | 0.0 | -1.5 | -2.6 | -3.0 | -2.6 |
(b) Plot these points and join with a smooth sine curve for \(0^\circ\le x\le 150^\circ\).
(c)(i) \(3\sin 2x+2=0\). This means \(3\sin 2x=-2\), so draw the line \(y=-2\) and read the crossing:
\[ \sin 2x=-\tfrac{2}{3} \Rightarrow 2x=221.8^\circ \Rightarrow x\approx 111^\circ \]Truth set \(=\{x\approx 111^\circ\}\).
(ii) \(\frac{3}{2}\sin 2x=0.25\). Multiply by 2: \(3\sin 2x=0.5\), so draw the line \(y=0.5\):
\[ \sin 2x=\tfrac{1}{6} \Rightarrow 2x=9.6^\circ \text{ or } 170.4^\circ \Rightarrow x\approx 5^\circ \text{ or } 85^\circ \]Truth set \(=\{x\approx 5^\circ,\; 85^\circ\}\).
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