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1.7 Trigonometry

1.7 Trigonometry

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Question 41

A landscape garden feature is designed using two identical sectors of a circle, ORP ORP\,ORP and OQSOQSOQS, and a central rhombus-shaped flower bed OSTROSTROSTR, as shown in the layout. The points PPP, OOO, and Q Q\,Q lie on a straight stone boundary of length 12 metres, such that O O\,O is the midpoint of PQPQPQ. The radii OR OR\,OR and OS OS\,OS form two sides of the rhombus, and the angle ∠ROS \angle ROS\,∠ROS is denoted by θ \theta\,θ radians.

Layout of two identical sectors ORP and OQS beside rhombus OSTR, with P, O and Q collinear, PQ equal to 12 metres, and angle ROS labelled theta.

a.

Show that the total area of the garden feature, A A\,A square metres, is given by

A=18(π−θ+2sin⁡θ) A = 18(\pi - \theta + 2\sin\theta) A=18(π−θ+2sinθ)
[4]
bi.

Use calculus to show that the maximum value of A A\,A occurs when θ=π3\displaystyle \theta = \frac{\pi}{3}θ=3π​. Fully justify that this value of θ \theta\,θ gives a maximum.

[5]
bii.

Determine the exact maximum value of AAA.

[2]
c.

Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if the total boundary length PQ PQ\,PQ were increased to 24 metres.

[2]
Markscheme

1.7 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.7 Trigonometry

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.

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