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**Deveno** $\sin\theta = \cos\theta = u$.

$\sin^2\theta + \cos^2\theta = 1$ (because these lie on the unit circle).

$2u^2 = 1$

$u = \pm \dfrac{\sqrt{2}}{2} = \pm a$

Thus the points on the unit circle where this occurs are: $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right), \left(-\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)$

If we form a triangle with the points $(0,0), (a,0) (a,a)$, we see it is an isosceles right-triangle, hence $\theta = \dfrac{\pi}{4} + 2k\pi$ (= 45 degrees (+360 degrees times an integer)).

Clearly, since $(a,a)$ and $(-a,-a)$ lie on the same line $y = x$ the angle between then is $\pi$ (180 degrees), so $(-a,-a)$ gives $\theta = \dfrac{5\pi}{4} + 2k\pi$

We can combine these as: $\theta = \dfrac{\pi}{4} + k\pi,\ k \in \Bbb Z$ <---the last summand is often omitted (except for $k = 1$, here), as we only bother to "go around the circle once".

A word about this: it assumes it is known that the sum of angles of an interior triangle sum to $\pi$ (or 180 degrees). This, actually, is a non-trivial assertion. HallsOfIvy's post assumes it is known that the radial length of 1/8-th of a circle is $\pi/4$. This, too, is also a non-trivial assertion (both length and angle are concepts that are harder to define than it might first appear).

It is unknown to me which non-trivial assertion you are assumed to "know" (it is unlikely you will see a "proof" of either of these facts in a trigonometry class; I presume the former, since it is possible you have been exposed to geometry, but perhaps unlikely you have been exposed to the tools (from calculus) necessary to compute arc-length). Nevertheless, both facts are "intuitively obvious", and you might be allowed to appeal to either one.