Find out $\theta$ when sin $\theta$'s and cos $\theta$'s value are given Given: $\sin \theta = \frac12$, $\cos \theta = \frac{\sqrt{3}}{2}$.
What I have tried: It is very easy looking at the angles' table and figuring out the value when the values of cos $\theta$ and sin $\theta$ are positive. But when either of them becomes negative, it becomes difficult for me to determine the value of $\theta$. Any idea how may I do it? Thanks. :)
[For example, $\sin{\theta}={\sqrt 3\over 2}$ and $\cos{\theta}=-{1\over 2}$ (note the negative sign).]
 A: In these kind of problems, it's easy to think of the unit circle. 
The $x$-coordinate represents the value of the cosine, the $y$-value represents the value of the sine.

Let's take $\sin(x)=\frac12\sqrt{3}$ and $\cos(x)=-\frac12$. Our $x$-value is negative and our $y$-value is positive. Therefore the angle we're looking for must lie in the second quadrant.
When we take a look at the unit circle, we immediately see that the angle is $\theta=\frac{2\pi}{3}$.
A: To work with the cases where the sine or cosine (or both) are negative, first drop the signs and determine the angle that you would have if everything were positive. The angle will be between $0$ and $\pi/2$ in that case.
Then you can make a drawing showing the radius of the unit circle at that angle in the first quadrant. Draw the reflection of this radius in the x-axis and the y-axis, and then draw the reflection of one of those in the other axis. These determine four related angles, one in each quadrant, with the property that their sines and cosines are the same up to sign. Pick the one that matches the signs you are given. (Remember that the cosine corresponds to the x-coordinate, and the sine corresponds to the y-coordinate.)
You know the angle the radius makes with the x-axis in  the first quadrant, so you know the complementary angle between the radius and the y-axis (it's $\pi/2$ minus the first angle). Theses angles are repeated within each of the quadrants, so just add them up as you travel around to reach the target point.
A: WARNING! Just to be on the safe side, make sure that the squares of your two data add up to 1; otherwise, something is wrong from the get-go.   
For a given value of $\cos{\theta}$ there is a secondary value for $\theta$.
For a given value of $\sin{\theta}$ there is a secondary value for $\theta$.   
To find these secondary values, use the formulæ below. The $\theta$ for which the values are equal is the $\theta$ you seek.   
For $\theta\in[0,2\pi)$:   

If $\sin{\theta}\ge 0$, the secondary value of $\theta$ is $\pi-\theta$.
  If $\sin{\theta}\lt 0$, the secondary value of $\theta$ is $3\pi-\theta$.   
For any $\cos{\theta}$, the secondary value of $\theta$ is $2\pi-\theta$.

