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).]
Best Answer
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 $\theta=\frac{2\pi}{3}$.