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 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.