Class 12 Inverse Trigonometric Functions · Long Answer3 Marksprincipal_values
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #101
Evaluate \(\sin[2\tan^{-1}(2/3)]+\cos(\tan^{-1}\sqrt3)\).
✅ Step-by-Step Solution
Final Answer: 37/26
- Let θ=arctan(2/3). The identity sin2θ=2tan θ/(1+tan²θ) applies.
- Thus sin2θ=(4/3)/(1+4/9)=(4/3)/(13/9)=12/13.
- The other term is cos(arctan√3)=cosπ/3=1/2.
- The sum is 12/13+1/2=(24+13)/26=37/26.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arctan(2/3). The identity sin2θ=2tan θ/(1+tan²θ) applies.
- [1 mark] Thus sin2θ=(4/3)/(1+4/9)=(4/3)/(13/9)=12/13.
- [1 mark] The other term is cos(arctan√3)=cosπ/3=1/2.
- [0.5 marks] The sum is 12/13+1/2=(24+13)/26=37/26.
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