To find the angle that completes the equation $\sec 40^\circ = \csc \theta$, we can use the relationship between secant and cosecant.

Since $\sec 40^\circ = \csc \theta$, we know that the angles $40^\circ$ and $\theta$ are complementary in this trigonometric identity. That is, they add up to $90^\circ$, because $\sec x = \csc (90^\circ - x)$ for complementary angles.

So, $\theta = 90^\circ - 40^\circ = 50^\circ$.

Answer: $\theta = 50^\circ$

Would you like more details on trigonometric identities or the steps?

Here are five related questions to delve deeper:

1. How do you find complementary angles in trigonometric functions?
2. Why do $\sec$ and $\csc$ relate to each other through complementary angles?
3. What are the other trigonometric identities for complementary angles?
4. How does this relationship apply to non-acute angles?
5. Can the identity $\sec x = \csc (90^\circ - x)$ be generalized?

Tip: Remember, complementary angles in trigonometry are always two angles that add up to $90^\circ$, which helps in solving similar equations involving secant and cosecant.