Class 12 Inverse Trigonometric Functions · Long Answer3 Marksbranch_sensitive_simplification

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #102

For \(-3\pi/4\le x\le\pi/4\), simplify \(\cos^{-1}[(3/5)\cos x+(4/5)\sin x]\).

✅ Step-by-Step Solution

Final Answer: Let φ=tan⁻¹(4/3). The value is 2π+x−φ for −3π/4≤x<φ−π, and φ−x for φ−π≤x≤π/4.

Let φ=arctan(4/3)∈(π/4,π/2), so cos φ=3/5 and sin φ=4/5. The argument equals cos(x−φ).
The stated interval makes u=x−φ range from −3π/4−φ (less than −π) up to π/4−φ (negative). It crosses −π at x=φ−π.
If x<φ−π, then −2π<u<−π and arccos(cos u)=2π+u. If x≥φ−π, then −π≤u<0 and arccos(cos u)=−u. Both give π at the joining point.
Thus the answer is 2π+x−φ on [−3π/4,φ−π), and φ−x on [φ−π,π/4].

[0.5 marks] Let φ=arctan(4/3)∈(π/4,π/2), so cos φ=3/5 and sin φ=4/5. The argument equals cos(x−φ).
[1 mark] The stated interval makes u=x−φ range from −3π/4−φ (less than −π) up to π/4−φ (negative). It crosses −π at x=φ−π.
[1 mark] If x<φ−π, then −2π<u<−π and arccos(cos u)=2π+u. If x≥φ−π, then −π≤u<0 and arccos(cos u)=−u. Both give π at the joining point.
[0.5 marks] Thus the answer is 2π+x−φ on [−3π/4,φ−π), and φ−x on [φ−π,π/4].

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