(i) $$ \sin 2 x \geq- an x $$ Toon aan dat $$ 2 \cos heta=\sin \left( heta+30^{\circ} ight) $$ geskryf kan word as $$ \sqrt{3} \sin heta=3 \cos heta $$ Los vervolgens die vergelyking $$ 2 \cos heta=\sin \left( heta+30^{\circ} ight) $$ op vir $$ heta \in\left[-180^{\circ} ; 180^{\circ} ight] $$. Teken die grafieke van die volgende funksies op diesclfde assestelsel $$ f( heta)=2 \cos heta $$ en $$ g( heta)=\sin \left( heta+30^{\circ} ight) $$ vir $$ heta \in\left[-180^{\circ} ight. $$. 1 ( $$ heta $$.

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(i) $$ \sin 2 x \geq-	an x $$ Toon aan dat $$ 2 \cos 	heta=\sin \left(	heta+30^{\circ}ight) $$ geskryf kan word as $$ \sqrt{3} \sin 	heta=3 \cos 	heta $$ Los vervolgens die vergelyking $$ 2 \cos 	heta=\sin \left(	heta+30^{\circ}ight) $$ op vir $$ 	heta \in\left[-180^{\circ} ; 180^{\circ}ight] $$. Teken die grafieke van die volgende funksies op diesclfde assestelsel $$ f(	heta)=2 \cos 	heta $$ en $$ g(	heta)=\sin \left(	heta+30^{\circ}ight) $$ vir $$ 	heta \in\left[-180^{\circ}ight. $$. 1 ( $$ 	heta $$.

UpStudy AI · Accepted Answer

It seems that I am unable to plot the functions directly. However, I can guide you on how to plot the functions $$ f(	heta) = 2 \cos 	heta $$ and $$ g(	heta) = \sin(	heta + 30^\circ) $$ using a graphing tool or software.

### Instructions for Plotting

1. **Choose a Graphing Tool**: You can use graphing calculators, software like Desmos, GeoGebra, or even Python with libraries like Matplotlib.

2. **Set the Range**: Make sure to set the x-axis (representing $$ 	heta $$) from $$-180^\circ$$ to $$180^\circ$$.

3. **Plot the Functions**:
   - For $$ f(	heta) = 2 \cos 	heta $$:
     - Input the function $$ y = 2 \cos(x) $$ where $$ x $$ is in degrees.
   - For $$ g(	heta) = \sin(	heta + 30^\circ) $$:
     - Input the function $$ y = \sin(x + 30) $$ where $$ x $$ is in degrees.

4. **Label the Axes**: Label the x-axis as $$ 	heta $$ and the y-axis as the function values.

5. **Identify Intersections**: Look for points where the two graphs intersect, which correspond to the solutions of the equation $$ 2 \cos 	heta = \sin(	heta + 30^\circ) $$.

### Summary of Solutions
The solutions to the equation $$ 2 \cos 	heta = \sin(	heta + 30^\circ) $$ in the interval $$ [-180^\circ, 180^\circ] $$ are:
- $$ 	heta = 60^\circ $$
- $$ 	heta = -120^\circ $$

If you have any further questions or need additional assistance, feel free to ask!
The solutions to the equation $$ 2 \cos 	heta = \sin(	heta + 30^\circ) $$ in the interval $$ [-180^\circ, 180^\circ] $$ are: - $$ 	heta = 60^\circ $$ - $$ 	heta = -120^\circ $$ To visualize these solutions, plot the functions $$ f(	heta) = 2 \cos 	heta $$ and $$ g(	heta) = \sin(	heta + 30^\circ) $$ on the same graph for $$ 	heta $$ from $$-180^\circ$$ to $$180^\circ$$. The points where the two graphs intersect correspond to the solutions $$ 	heta = 60^\circ $$ and $$ 	heta = -120^\circ $$.

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