Key Concepts and Formulas

• Modulus of a complex number: For $z = x + iy$, the modulus is $r = |z| = \sqrt{x^2 + y^2}$.
• Argument of a complex number: The argument $\theta$ satisfies $\tan \theta = \frac{y}{x}$. The quadrant of $z$ determines the correct value of $\theta$.
• Trigonometric Identities: $1 + \tan^2 A = \sec^2 A$ and $\sec(\pi - A) = -\sec A$.

Step 1: Identify the Real and Imaginary Parts

We are given the complex number $z = 2 - i(2 \tan \frac{5\pi}{8})$. We need to identify the real and imaginary parts, $x$ and $y$, respectively.

• Real part: $x = 2$
• Imaginary part: $y = -2 \tan \frac{5\pi}{8}$

Step 2: Determine the Quadrant of the Complex Number

To find the correct argument, we must identify the quadrant in which the complex number lies. The angle $\frac{5\pi}{8}$ is in the second quadrant, where the tangent function is negative. Therefore, $\tan \frac{5\pi}{8} < 0$. Thus, $y = -2 \tan \frac{5\pi}{8}$ is positive. Since $x = 2$ is also positive, the complex number $z$ lies in the first quadrant.

Step 3: Calculate the Modulus, $r$

We use the formula $r = \sqrt{x^2 + y^2}$ to calculate the modulus. $r = \sqrt{2^2 + \left(-2 \tan \frac{5\pi}{8}\right)^2} = \sqrt{4 + 4 \tan^2 \frac{5\pi}{8}} = \sqrt{4(1 + \tan^2 \frac{5\pi}{8})}$ Using the trigonometric identity $1 + \tan^2 A = \sec^2 A$, we get $r = \sqrt{4 \sec^2 \frac{5\pi}{8}} = 2 \left| \sec \frac{5\pi}{8} \right|$ Since $\frac{5\pi}{8}$ is in the second quadrant, $\cos \frac{5\pi}{8} < 0$, and therefore $\sec \frac{5\pi}{8} < 0$. Thus, $\left| \sec \frac{5\pi}{8} \right| = - \sec \frac{5\pi}{8}$. $r = -2 \sec \frac{5\pi}{8}$ Now, we use the identity $\sec(\pi - A) = -\sec A$. Since $\frac{5\pi}{8} = \pi - \frac{3\pi}{8}$, we have $\sec \frac{5\pi}{8} = \sec(\pi - \frac{3\pi}{8}) = -\sec \frac{3\pi}{8}$. Therefore, $r = -2 \left( -\sec \frac{3\pi}{8} \right) = 2 \sec \frac{3\pi}{8}$

Step 4: Calculate the Argument, $\theta$

We use the formula $\tan \theta = \frac{y}{x}$ to find the argument. $\tan \theta = \frac{-2 \tan \frac{5\pi}{8}}{2} = -\tan \frac{5\pi}{8}$ Since $z$ is in the first quadrant, we seek an angle $\theta$ in $(0, \frac{\pi}{2})$ such that $\tan \theta = -\tan \frac{5\pi}{8}$. Using the identity $\tan(\pi - A) = -\tan A$, we have $-\tan \frac{5\pi}{8} = \tan(\pi - \frac{5\pi}{8}) = \tan \frac{3\pi}{8}$. Thus, $\tan \theta = \tan \frac{3\pi}{8}$. Since $\frac{3\pi}{8}$ is in the first quadrant, we have $\theta = \frac{3\pi}{8}$.

Step 5: Combine the Modulus and Argument

Therefore, the complex number $z$ has modulus $r = 2 \sec \frac{3\pi}{8}$ and argument $\theta = \frac{3\pi}{8}$. Thus, $(r, \theta) = \left(2 \sec \frac{3\pi}{8}, \frac{3\pi}{8}\right)$.

Common Mistakes & Tips to Avoid

• Sign of Modulus: Always ensure the modulus is positive. When simplifying expressions involving square roots, be careful to consider the sign of the expression inside the absolute value.
• Quadrant of Argument: Always determine the quadrant of the complex number to find the correct argument. The arctangent function only returns values in $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Summary

We found the modulus and argument of the given complex number by first identifying its real and imaginary parts. We then calculated the modulus using the formula $r = \sqrt{x^2 + y^2}$, carefully considering the sign of the secant function. Next, we found the argument using the formula $\tan \theta = \frac{y}{x}$, ensuring that $\theta$ was in the correct quadrant. The final result is $(r, \theta) = \left(2 \sec \frac{3\pi}{8}, \frac{3\pi}{8}\right)$.

The final answer is \boxed{\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)}, which corresponds to option (B).