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 (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.