Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #66
Prove \(\cos^{-1}(4/5)+\cos^{-1}(12/13)=\cos^{-1}(33/65)\).
✅ Step-by-Step Solution
Final Answer: Identity proved.
- Let α=arccos(4/5), β=arccos(12/13), both acute. Then sin α=3/5 and sin β=5/13.
- Apply cos(α+β)=cos α cos β−sin α sin β=(4·12−3·5)/65=33/65.
- The sum of two acute angles lies in (0,π), exactly inside the arccosine principal interval.
- Thus α+β=arccos(33/65), proving the statement.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let α=arccos(4/5), β=arccos(12/13), both acute. Then sin α=3/5 and sin β=5/13.
- [1 mark] Apply cos(α+β)=cos α cos β−sin α sin β=(4·12−3·5)/65=33/65.
- [1 mark] The sum of two acute angles lies in (0,π), exactly inside the arccosine principal interval.
- [0.5 marks] Thus α+β=arccos(33/65), proving the statement.
Master Chapter 2: Inverse Trigonometric Functions
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Inverse Trigonometric Functions.
← Previous Question
Prove \(\sin^{-1}(63/65)=\sin^{-1}(5/13)+\cos^{-1}(3/5)\).
Next Question →Solve \(\sin^{-1}(1-x)-2\sin^{-1}x=\pi/2\).
More Practice Questions in Inverse Trigonometric Functions
Related Questions in Inverse Trigonometric Functions
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}(\cos(13\pi/5))\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Find the full real solution set of \(\sin^{-1}x>\cos^{-1}x\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin[\pi/2-\sin^{-1}(-1/2)]\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For \(x\in[-1,1]\setminus\{0\}\), simplify \(\cot(\sin^{-1}x)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}(\cos(\pi/9))\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\theta=\sin^{-1}(\sin(-600^\circ))\), giving the answer in radians.