Question
If tan(α + iβ) = i, where α and β are real then α will be _____.
The correct answer is
No finite real values of α and β satisfy the equation.
For a finite complex z, \(\tan z=-i\frac{e^{2iz}-1}{e^{2iz}+1}\). Setting it equal to i would force \(e^{2iz}=0\), impossible for an exponential. Thus no finite real α,β satisfy the equation. The issue is nonexistence, not uncertainty.
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Important Questions from Inverse Trigonometric Functions
Solve for x?
sin 2x - 4 sin x + 3 = 0, 0 ≤ x ≤ \(π/2\)
What is the principal solutions of the equation \(\tan x=-\frac{1}{\sqrt{3}}\) ?
- If \(4 \sin^{-1} x + \cos^{-1} x = \pi\), then what is \(\sin^{-1} x + 4 \cos^{-1} x\) equal to?
- What is \(\cot^2 (\sec^{-1}2) + \tan^2 (\text{cosec}^{-1}3)\) equal to?
- Consider the following in respect of inverse circular functions :
I. \(\sin^{-1}(-x) = -\sin^{-1} x\)
II. \(\cos^{-1}(-x) = \cos^{-1} x\)
III. \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
IV. \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
How many of the above are correct ?
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