A motion-activated security sensor at a warehouse entrance sweeps a circular sector OABOABOAB with centre OOO. The sensor has a detection range of r=106 mr = 10\sqrt{6}\text{ m}r=106 m and its field of vision is defined by the central angle AOB=π6AOB = \frac{\pi}{6}AOB=6π radians. A straight security barrier is installed on the floor connecting points AAA and BBB, creating a segment-shaped region between the barrier and the arc ABABAB.
Show that the area of this segment-shaped region is
k(π−3) m2 k(\pi - 3) \text{ m}^2 k(π−3) m2where kkk is an integer to be found.
317 exam-style questions on AQA A Level Maths 1.8 E: Trigonometry, covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context. Each one has a worked solution and a mark scheme showing where the marks go.