1. **State the problem:** We want to express the complex number $1 \pm \frac{i}{2}$ in the form $\frac{\sqrt{5}}{2}(\cos\varphi + i\sin\varphi)$ and find the angle $\varphi$. 2. **Recall the formula:** A complex number $z = x + iy$ can be written in polar form as $r(\cos\varphi + i\sin\varphi)$ where $r = \sqrt{x^2 + y^2}$ and $\varphi = \arctan\frac{y}{x}$. 3. **Calculate the modulus $r$:** $$r = \sqrt{1^2 + \left(\pm \frac{1}{2}\right)^2} = \sqrt{1 + \frac{1}{4}} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}$$ 4. **Calculate the argument $\varphi$:** For $1 + \frac{i}{2}$: $$\varphi = \arctan\left(\frac{\frac{1}{2}}{1}\right) = \arctan\left(\frac{1}{2}\right)$$ For $1 - \frac{i}{2}$: $$\varphi = \arctan\left(\frac{-\frac{1}{2}}{1}\right) = \arctan\left(-\frac{1}{2}\right)$$ 5. **Final expression:** $$1 \pm \frac{i}{2} = \frac{\sqrt{5}}{2} \left( \cos\varphi + i \sin\varphi \right)$$ where $$\varphi = \arctan\left(\pm \frac{1}{2}\right)$$ This matches the given form exactly. **Answer:** $$\varphi = \arctan\left(\pm \frac{1}{2}\right)$$