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Question #41158

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_k = \left( \frac{1 + i \tan \frac{\pi}{18}}{1 - i \tan \frac{\pi}{18}} \right)^k$ for $k \in \mathbb{N}$ . Find the value of the following expression in degrees: $\cos^{-1}(\text{Re}(z_{13})) + \sin^{-1}(\text{Im}(z_{10})) + \tan^{-1}\left(\frac{\text{Im}(z_{16})}{\text{Re}(z_{16})}\right)$

Options

A.
$40^\circ$
B.
$160^\circ$
C.
$20^\circ$
D.
$780^\circ$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Principal Value Branch Boundaries|Algebraic Manipulation of i|Radian-Degree Conversion Identity

Why This Question Is Tricky

Common Mistakes

This question forces a collision between complex number algebra and the geometric constraints of inverse trigonometric functions.

The student must first use Euler's formula to reduce the complex ratio into a standard polar form.

The primary trap lies in the blind application of the identity $f^{-1}(f(x)) = x$ .

Students who ignore the principal value branches of ITF will incorrectly sum the raw arguments ( $260^\circ + 200^\circ + 320^\circ = 780^\circ$ ) or make sign errors when reducing to the first quadrant (e.g., yielding $160^\circ$ ).

It rigorously tests domain awareness.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Principal Value Branch Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free