Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>Which of the following is/are correct?</p><p>(a) <math>\cos(\cos(\cos^{-1}1)) < \sin(\sin^{-1}(\sin(\pi -1))) < \sin(\cos^{-1}(\cos(2\pi -2)))</math></p><p>(b) <math>\cos(\cos(\cos^{-1}1)) < \sin(\cos^{-1}(\cos(2\pi -2))) < \sin(\sin^{-1}(\sin(\pi -1))) < \tan(\cot^{-1}(\cot 1))</math></p><p>(c) <math>\sum_{t=1}^{5000} \cos^{-1}(\cos(2t\pi -1)) = \sum_{t=1}^{2500} \cot^{-1}(\cot(t\pi + 2))</math>, where <math>t \in \mathbb{Z}</math></p><p>(d) <math>\cot^{-1}\cot\left(\cosec^{-1}\cosec\right)</math></p>
<p>(a) <math>\cos(\cos(\cos^{-1}1)) < \sin(\sin^{-1}(\sin(\pi -1))) < \sin(\cos^{-1}(\cos(2\pi -2)))</math></p>
<p>(b) <math>\cos(\cos(\cos^{-1}1)) < \sin(\cos^{-1}(\cos(2\pi -2))) < \sin(\sin^{-1}(\sin(\pi -1))) < \tan(\cot^{-1}(\cot 1))</math></p>
<p>(c) <math>\sum_{t=1}^{5000} \cos^{-1}(\cos(2t\pi -1)) = \sum_{t=1}^{2500} \cot^{-1}(\cot(t\pi + 2))</math>, where <math>t \in \mathbb{Z}</math></p>
<p>(d) <math>\cot^{-1}\cot\left(\cosec^{-1}\cosec\right)</math></p>
Step-by-Step Solution
Key Concept: Understanding the ranges and properties of inverse trigonometric functions is essential to compare and evaluate composite expressions correctly.
<p><strong>Solution:</strong></p><p>For (a): Evaluate each term using properties of inverse trigonometric functions.</p><p>For (b): Compare the bounds carefully using range restrictions of inverse functions.</p><p>For (c): Use periodicity and summation properties to verify the equality.</p>
Correct Answer: A, B, C