A landscape architect is designing a fan-shaped flower bed which is a sector of a circle with radius rrr metres and central angle ϕ\phiϕ radians. The design requires the flower bed to have a total perimeter of 18 m18\text{ m}18 m and an area of 20 m220\text{ m}^220 m2.
Show that 5ϕ2−21ϕ+18=05\phi^2 - 21\phi + 18 = 05ϕ2−21ϕ+18=0 is NOT correct for these dimensions, and instead determine the correct quadratic equation in terms of ϕ\phiϕ only, in the form aϕ2+bϕ+c=0a\phi^2 + b\phi + c = 0aϕ2+bϕ+c=0, where a,b,ca, b, ca,b,c are integers.
Hence find the two possible pairs of values for rrr and ϕ\phiϕ.
317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.