(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\(^o\)...

(a) Copy and complete the table of values for the relation y = 3 sin 2x. 

(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 of Values for $\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">y</span>}<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">=</span>\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">3</span>}\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">sin</span>}<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;"></span>\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">2</span>}\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">x</span>}$y = 3 \sin 2xsinx

To complete the table, we need to calculate the values of $yy$ for given $xx$ values using the equation $y=3\sin 2xy\; =\; 3\; \backslash sin\; 2x$.

Let's calculate the values for each $xx$:

• $x=0^{\circ }x\; =\; 0^\backslash circ$

  $y=3\sin 2(0^{\circ })=3\sin 0=0y\; =\; 3\; \backslash sin\; 2(0^\backslash circ)\; =\; 3\; \backslash sin\; 0\; =\; 0$

• $x=15^{\circ }x\; =\; 15^\backslash circ$

  $y=3\sin 2(15^{\circ })=3\sin 30^{\circ }=3\times 0.5=1.5y\; =\; 3\; \backslash sin\; 2(15^\backslash circ)\; =\; 3\; \backslash sin\; 30^\backslash circ\; =\; 3\; \backslash times\; 0.5\; =\; 1.5$

• $x=30^{\circ }x\; =\; 30^\backslash circ$

  $y=3\sin 2(30^{\circ })=3\sin 60^{\circ }=3\times \frac{\sqrt{3}}{2}\approx 2.598y\; =\; 3\; \backslash sin\; 2(30^\backslash circ)\; =\; 3\; \backslash sin\; 60^\backslash circ\; =\; 3\; \backslash times\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; \backslash approx\; 2.598$

• $x=45^{\circ }x\; =\; 45^\backslash circ$

  $y=3\sin 2(45^{\circ })=3\sin 90^{\circ }=3\times 1=3y\; =\; 3\; \backslash sin\; 2(45^\backslash circ)\; =\; 3\; \backslash sin\; 90^\backslash circ\; =\; 3\; \backslash times\; 1\; =\; 3$

• $x=60^{\circ }x\; =\; 60^\backslash circ$

  $y=3\sin 2(60^{\circ })=3\sin 120^{\circ }=3\times \frac{\sqrt{3}}{2}\approx 2.598y\; =\; 3\; \backslash sin\; 2(60^\backslash circ)\; =\; 3\; \backslash sin\; 120^\backslash circ\; =\; 3\; \backslash times\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; \backslash approx\; 2.598$

• $x=75^{\circ }x\; =\; 75^\backslash circ$

  $y=3\sin 2(75^{\circ })=3\sin 150^{\circ }=3\times 0.5=1.5y\; =\; 3\; \backslash sin\; 2(75^\backslash circ)\; =\; 3\; \backslash sin\; 150^\backslash circ\; =\; 3\; \backslash times\; 0.5\; =\; 1.5$

• $x=90^{\circ }x\; =\; 90^\backslash circ$

  $y=3\sin 2(90^{\circ })=3\sin 180^{\circ }=3\times 0=0y\; =\; 3\; \backslash sin\; 2(90^\backslash circ)\; =\; 3\; \backslash sin\; 180^\backslash circ\; =\; 3\; \backslash times\; 0\; =\; 0$

• $x=105^{\circ }x\; =\; 105^\backslash circ$

  $y=3\sin 2(105^{\circ })=3\sin 210^{\circ }=3\times (-0.5)=-1.5y\; =\; 3\; \backslash sin\; 2(105^\backslash circ)\; =\; 3\; \backslash sin\; 210^\backslash circ\; =\; 3\; \backslash times\; (-0.5)\; =\; -1.5$

• $x=120^{\circ }x\; =\; 120^\backslash circ$

  $y=3\sin 2(120^{\circ })=3\sin 240^{\circ }=3\times (-\frac{\sqrt{3}}{2})\approx -2.598y\; =\; 3\; \backslash sin\; 2(120^\backslash circ)\; =\; 3\; \backslash sin\; 240^\backslash circ\; =\; 3\; \backslash times\; (-\backslash frac\{\backslash sqrt\{3\}\}\{2\})\; \backslash approx\; -2.598$

• $x=135^{\circ }x\; =\; 135^\backslash circ$

  $y=3\sin 2(135^{\circ })=3\sin 270^{\circ }=3\times (-1)=-3y\; =\; 3\; \backslash sin\; 2(135^\backslash circ)\; =\; 3\; \backslash sin\; 270^\backslash circ\; =\; 3\; \backslash times\; (-1)\; =\; -3$

• $x=150^{\circ }x\; =\; 150^\backslash circ$

  $y=3\sin 2(150^{\circ })=3\sin 300^{\circ }=3\times (-\frac{\sqrt{3}}{2})\approx -2.598y\; =\; 3\; \backslash sin\; 2(150^\backslash circ)\; =\; 3\; \backslash sin\; 300^\backslash circ\; =\; 3\; \backslash times\; (-\backslash frac\{\backslash sqrt\{3\}\}\{2\})\; \backslash approx\; -2.598$

Here is the completed table:

b)

c)

i) 113

(ii) 3o ${}^{\mathrm{<br>}}$ and 82o