Verified answer

Note first that sqrt(1^2 + (sqrt(3))^2) = sqrt(1 + 3) = 2.

So if we write the expression as

2*[(1/2)*sin(theta) + (sqrt(3) /2)*cos(theta)], then if

we have an angle a such that cos(a) = (1/2) and

sin(a) = sqrt(3) /2 then the expression becomes

2*[cos(a)*sin(theta) + sin(a)*cos(theta)] = 2*sin(theta + a).

Now we know that there exists a value a that satisfies

cos(a) = 1/2 and sin(a) = sqrt(3) /2 such that 0 < a < 90

since both values are positive and sin^2(a) + cos^2(a) = 1.

Now if cos(a) = 1/2, then for 0 < a < 90 we have a = 60 degrees.

So the expression is 2*sin(theta + 60).

Edit: Now for the second part.

If we have 2*sin(theta + 60) = 1 then sin(theta + 60) = 1/2,

so either (theta + 60) = 30 + n*360 or

(theta + 60) = 150 + n*360 for any integer n.

Thus theta = -30 + n*360 or theta = 90 + n*360,

which for 0 < theta < 360 we have theta = 330 degrees or

theta = 90 degrees.

Hope I gave a better explanation this time. :)