Key Concepts and Formulas

• Complex Number Manipulation: To simplify a complex number of the form $\frac{1}{a+ib}$, we multiply the numerator and denominator by the conjugate of the denominator ($a-ib$) to express it in the standard form $x+iy$. The real part of $x+iy$ is $x$.
• Trigonometric Identities: We will use the half-angle formulas for cosine and sine:
  • $1 - \cos \theta = 2\sin^2(\frac{\theta}{2})$
  • $\sin \theta = 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})$
• Definite Integration: The integral of $\sin x$ is $-\cos x$. We will evaluate the definite integral using the Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is an antiderivative of $f(x)$.

Step-by-Step Solution

Step 1: Simplify the complex number and find its real part. We are given the complex number $z = (1 - \cos \theta + 2i\sin \theta )^{ - 1}$. To find its real part, we first rewrite it as: $z = \frac{1}{1 - \cos \theta + 2i\sin \theta}$ Now, we use the trigonometric identities $1 - \cos \theta = 2\sin^2(\frac{\theta}{2})$ and $\sin \theta = 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})$ to substitute into the expression: $z = \frac{1}{2\sin^2(\frac{\theta}{2}) + 2i(2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2}))}$ $z = \frac{1}{2\sin^2(\frac{\theta}{2}) + 4i\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})}$ To simplify further, we can factor out $2\sin(\frac{\theta}{2})$ from the denominator: $z = \frac{1}{2\sin(\frac{\theta}{2}) (\sin(\frac{\theta}{2}) + 2i\cos(\frac{\theta}{2}))}$ Now, to find the real part, we multiply the numerator and denominator by the conjugate of the term in the parenthesis, which is $(\sin(\frac{\theta}{2}) - 2i\cos(\frac{\theta}{2}))$: $z = \frac{1}{2\sin(\frac{\theta}{2})} \times \frac{\sin(\frac{\theta}{2}) - 2i\cos(\frac{\theta}{2})}{(\sin(\frac{\theta}{2}) + 2i\cos(\frac{\theta}{2}))(\sin(\frac{\theta}{2}) - 2i\cos(\frac{\theta}{2}))}$ The denominator becomes: $(\sin(\frac{\theta}{2}))^2 - (2i\cos(\frac{\theta}{2}))^2 = \sin^2(\frac{\theta}{2}) - (4i^2\cos^2(\frac{\theta}{2}))$ Since $i^2 = -1$: $= \sin^2(\frac{\theta}{2}) - (-4\cos^2(\frac{\theta}{2})) = \sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2})$ So, the complex number $z$ is: $z = \frac{1}{2\sin(\frac{\theta}{2})} \times \frac{\sin(\frac{\theta}{2}) - 2i\cos(\frac{\theta}{2})}{\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2})}$ $z = \frac{\sin(\frac{\theta}{2})}{2\sin(\frac{\theta}{2})(\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2}))} - i \frac{2\cos(\frac{\theta}{2})}{2\sin(\frac{\theta}{2})(\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2}))}$ The real part of $z$ is: $\text{Re}(z) = \frac{\sin(\frac{\theta}{2})}{2\sin(\frac{\theta}{2})(\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2}))} = \frac{1}{2(\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2}))}$ We are given that the real part is $\frac{1}{5}$: $\frac{1}{2(\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2}))} = \frac{1}{5}$ This implies: $2(\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2})) = 5$ $\sin^2(\frac{\theta}{2}) + 4\cos^2(\frac{\theta}{2}) = \frac{5}{2}$ We can rewrite $\sin^2(\frac{\theta}{2})$ as $1 - \cos^2(\frac{\theta}{2})$: $(1 - \cos^2(\frac{\theta}{2})) + 4\cos^2(\frac{\theta}{2}) = \frac{5}{2}$ $1 + 3\cos^2(\frac{\theta}{2}) = \frac{5}{2}$ $3\cos^2(\frac{\theta}{2}) = \frac{5}{2} - 1 = \frac{3}{2}$ $\cos^2(\frac{\theta}{2}) = \frac{1}{2}$ This means $\cos(\frac{\theta}{2}) = \pm \frac{1}{\sqrt{2}}$.

Step 2: Determine the value of $\theta$. We are given that $\theta \in (0, \pi)$. This implies that $\frac{\theta}{2} \in (0, \frac{\pi}{2})$. In this interval, $\cos(\frac{\theta}{2})$ is positive. Therefore, we take the positive root: $\cos(\frac{\theta}{2}) = \frac{1}{\sqrt{2}}$ For $\frac{\theta}{2} \in (0, \frac{\pi}{2})$, this implies: $\frac{\theta}{2} = \frac{\pi}{4}$ So, $\theta = \frac{\pi}{2}$.

Step 3: Evaluate the definite integral. We need to find the value of the integral $\int_0^\theta {\sin x} dx$. Substitute the value of $\theta$ we found: $\int_0^{\pi/2} {\sin x} dx$ The antiderivative of $\sin x$ is $-\cos x$. Evaluating the definite integral: $[-\cos x]_0^{\pi/2} = -\cos(\frac{\pi}{2}) - (-\cos(0))$ $= -0 - (-1)$ $= 1$

Common Mistakes & Tips

• Trigonometric Simplification: Ensure accurate application of half-angle identities. Mistakes here will lead to an incorrect value for $\theta$.
• Complex Conjugate: When finding the real part of a fraction, always multiply by the conjugate of the denominator. Forgetting this or miscalculating the conjugate will lead to errors.
• Range of $\theta$: Pay close attention to the given range of $\theta$ to select the correct sign for trigonometric functions of $\frac{\theta}{2}$.

Summary

The problem requires us to first simplify a given complex number using trigonometric identities and the concept of complex conjugates to find the real part. By equating this real part to the given value $\frac{1}{5}$, we determined the specific value of the angle $\theta$. Finally, we evaluated the definite integral of $\sin x$ from $0$ to this determined value of $\theta$.

The final answer is \boxed{1}.