<s>[INST] How to simplify this trignometric expression: $4( 3 \sin \theta)( 3 \cos \theta)$? I was given a circle with a radius of $3$ and in it was a rectangle and an angle $\theta$ extending from the $x$ axis to up with coordinates of $(3 \cos \theta, 3 \sin \theta)$ and the question asks me to show that the area of the triangle represented by $A$ is equal to $18 \sin 2 \theta$. I figured that the the rectangle has $8$ triangles so using the angle and the coordinates I would find the area of one triangle and multiple it by $8$. So I ended up with $$(8)(.5) b h= (8)(.5)(3 \cos \theta) (3 \sin\theta) = 4( 3 \sin \theta)( 3 \cos \theta).$$ How do you represent this as $18 \sin 2 \theta$?
Thanks and sorry for the question being too long.
 [/INST] $\text{Area}= \frac 1 2 3\sin\theta 3\cos\theta$
Use $\sin\theta \cos\theta = \frac 1 2\sin 2\theta$ 
$=\frac 9 4 \sin 2\theta$
Multiply by $8$ to get $18\sin 2\theta$
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