1. The problem asks for the "primary capacity" of the complex number $\cos(50) + i \sin(50) + i$. 2. First, recognize that $\cos(50) + i \sin(50)$ is in the form of Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ with $\theta = 50$ degrees. 3. The complex number can be rewritten as $\cos(50) + i(\sin(50) + 1)$. 4. To find the magnitude (or modulus) of the complex number, use the formula: $$|z| = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2}$$ 5. Substitute the values: $$|z| = \sqrt{\cos^2(50) + (\sin(50) + 1)^2}$$ 6. Calculate each term: $$\cos^2(50) = (\cos(50))^2$$ $$ (\sin(50) + 1)^2 = \sin^2(50) + 2\sin(50) + 1$$ 7. Sum the terms inside the square root: $$|z| = \sqrt{\cos^2(50) + \sin^2(50) + 2\sin(50) + 1}$$ 8. Use the Pythagorean identity $\cos^2(\theta) + \sin^2(\theta) = 1$: $$|z| = \sqrt{1 + 2\sin(50) + 1} = \sqrt{2 + 2\sin(50)}$$ 9. Factor out 2: $$|z| = \sqrt{2(1 + \sin(50))} = \sqrt{2} \sqrt{1 + \sin(50)}$$ 10. This is the magnitude or "primary capacity" of the complex number. Final answer: $$|z| = \sqrt{2} \sqrt{1 + \sin(50)}$$