To solve these problems, we need to use trigonometric identities and properties of angles in different quadrants. Let's go through each part systematically.

Problem Analysis and Solutions:

1. State all angles between $0^\circ$ and $360^\circ$ that make each statement true:

   a) $\sin 200^\circ = \sin \underline{\ \ \ }$

   • To find the other angle with the same sine value, we use the fact that sine is positive in the first and second quadrants and negative in the third and fourth quadrants.
   • The reference angle for $200^\circ$ is $200^\circ - 180^\circ = 20^\circ$.
   • So, the other angle with the same sine is $360^\circ - 200^\circ = 160^\circ$.
   • Answer: $\sin 200^\circ = \sin 160^\circ$.

   b) $\tan \theta = -2.747 \Rightarrow \theta = \underline{\ \ \ \ } \text{ or } \underline{\ \ \ \ }$

   • Tangent is negative in the second and fourth quadrants.
   • Using the arctangent of $2.747$, we find the reference angle as $\tan^{-1}(2.747) \approx 70^\circ$.
   • The solutions in the given range are:
     • $180^\circ - 70^\circ = 110^\circ$ (second quadrant).
     • $360^\circ - 70^\circ = 290^\circ$ (fourth quadrant).
   • Answer: $\theta = 110^\circ$ or $\theta = 290^\circ$.

   c) $\cos 135^\circ = -\cos \underline{\ \ \ \ } \text{ or } \underline{\ \ \ \ }$

   • The angle $135^\circ$ is in the second quadrant, where cosine is negative.
   • We want an angle with the same reference angle where cosine is positive.
   • The reference angle for $135^\circ$ is $135^\circ - 90^\circ = 45^\circ$.
   • The other angles where cosine has the same magnitude but is positive are:
     • $45^\circ$ and $360^\circ - 45^\circ = 315^\circ$.
   • Answer: $\cos 135^\circ = -\cos 45^\circ$ or $-\cos 315^\circ$.

   d) $\tan(-100^\circ) = \tan \underline{\ \ \ \ } \text{ or } \underline{\ \ \ \ }$

   • To find an equivalent positive angle for $-100^\circ$, add $360^\circ$:
     • $-100^\circ + 360^\circ = 260^\circ$.
   • The other angle with the same tangent value in the given range is:
     • $180^\circ + 80^\circ = 80^\circ$.