Around and around we go

The expression reminds us of $\sin 2\theta$ which is $2\times \sin \theta \cos \theta$ . We multiply both sides of the expression by two to come up with the neat expression of $\sin 2\theta = 1$ . Sine is 1 when the angle is 90 or co terminal to 90. The range of $2\theta$ is 0 to 2,000. Thus we have 90, 450, 810, 1170, 1530, and 1890 or $\boxed {6}$ angles.

sinAcosA = 1/2 only when A = 45 deg so A = {45,225,405,585,765,945} so A has 6 values.

Note that $\sin\theta \cos\theta=\frac{2\sin\theta\cos\theta}{2}=\frac{\sin2\theta}{2}=\frac{1}{2}$ . So we have $\sin2\theta=1$ and $\theta=\frac{(2k-1)\pi}{2}$ for $k\in\mathbb{Z}$ . The smallest solution that falls in the range $[0^{\circ},1000^{\circ}]$ is for $k=1$ and the largest is for $k=6$ . So there are $\boxed{6}$ solutions.

take 2 to left hand side... this makes \sin \2theta=1 hence we get \theta=pi/4 for solutions till 1000 degrees add pi to pi/4..

Doing some Algebra manipulation we get:

$2\sin \theta\cos \theta = 1 \rightarrow \sin 2\theta = 1$

We know that if $\theta = 90, 270$ then the equation is true.

So the general solution is $45 + 180n$ will be the general solution.

$0 \le45 + 180n < 1000$ Solving for nonegative integers we get:

$0\le n \le 5$

Thus, $\boxed{6}$ possibilities.

$\sin\theta\cos\theta = \frac{1}{2}$ simplifies into:
$2\sin\theta\cos\theta = 1$
Using the trig identity: $2\sin\theta\cos\theta = \sin2\theta = 1$
$\sin2\theta = 1$ when $2\theta= 90 degrees + 360 degrees$
This happens when $\theta= 45 degrees+180 degrees$
The solutions are the numbers: 45, 225, 405, 585, 765, and 945 degrees.
This means there are $\boxed{6 solutions.}$