Question

The principal argument of the complex number $z=\frac{1+\sin\pi-i\cos\pi}{1+\sin\pi+i\cos\pi}$ is

• A. $\frac{\pi}{3}$
• B. $\frac{\pi}{6}$
• C. $\frac{\pi}{5}$
• D. $\frac{\pi}{2}$
• E. $\frac{\pi}{4}$

Hint:

Trig Tip: Always check if expressions involve exact, easy-to-evaluate angles like $0, \pi/2$, or $\pi$ before trying to apply complex identities. Simplifying early saves massive amounts of algebraic work.

Answer 1

The Correct Option is D

Concept:
The principal argument of a complex number $z = x + iy$, denoted as $\text{Arg}(z)$, is the angle $\theta$ formed with the positive real axis, such that $-\pi < \theta \le \pi$. To find it, first simplify the complex fraction into standard Cartesian form $x + iy$ by evaluating trigonometric terms and rationalizing the denominator.

Step 1: Evaluate the trigonometric values.
Recall the standard values for sine and cosine at $\pi$ radians ($180^{\circ}$): $$\sin \pi = 0$$ $$\cos \pi = -1$$

Step 2: Substitute these values into the given equation.
Replace the trigonometric functions in both the numerator and the denominator with their numerical values: Numerator: $1 + (0) - i(-1) = 1 + i$ Denominator: $1 + (0) + i(-1) = 1 - i$

Step 3: Rewrite the simplified complex fraction.
The complex number expression simplifies significantly to: $$z = \frac{1 + i}{1 - i}$$

Step 4: Rationalize the denominator.
Multiply the numerator and the denominator by the complex conjugate of the denominator, which is $(1 + i)$: $$z = \frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}$$ $$z = \frac{1 + i + i + i^2}{1^2 - i^2}$$ Since $i^2 = -1$: $$z = \frac{1 + 2i - 1}{1 - (-1)} = \frac{2i}{2} = i$$

Step 5: Determine the principal argument.
The simplified complex number is $z = 0 + 1i$. This point lies strictly on the positive imaginary axis. The angle from the positive real axis to the positive imaginary axis is exactly $90^{\circ}$, or $\frac{\pi}{2}$ radians. Therefore, $\text{Arg}(z) = \frac{\pi}{2}$. Hence the correct answer is (D) $\frac{\pi{2}$}.