Maths · Trigonometrical identities and trigonometrical functions
In a the angles and are two different values of satisfying The triangle:
In a \( \triangle A B C, \) the angles \( A \) and \( B \) are two different values of \( \theta \) satisfying \( \sqrt{\mathbf{3}} \cos \theta+\sin \theta=k,|k|<2 . \) The triangle:
- A. is an acute angled
- B. is aright angled
- C. is an obtuse angled
- D. has one angle \( =\frac{\pi}{3} \)
Step-by-step solution
The equation √3 cos θ + sin θ = k can be rewritten as 2 cos(θ - π/6) = k, so cos(θ - π/6) = k/2. Since |k| < 2, there are two distinct solutions for θ in (0, π) only when k > √3, leading to α = arccos(k/2) ∈ (0, π/6). Then the two angles are A = π/6 + α and B = π/6 - α, so A + B = π/3. Hence C = π - (A+B) = 2π/3, which is obtuse. Thus triangle is obtuse angled.
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