Answer
There is no angle $\theta$ whose secant value is $\frac{1}{2}$ because the value of $\sec{\theta}$ will never be less than $1$.
Work Step by Step
Definition I states that $\sec{\theta}= \dfrac{r}{x}, x\ne0$.
Figure 1 shows that $r$ is the hypotenuse (longest side) of the right triangle, while $x$ and $y$ are the legs.
Since $r$ is longer than $x$, $\dfrac{r}{x}$ will never be smaller than $1$.
Thus, there will be no angle $\theta$ whose secant value is $\frac{1}{2}$.
