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1.7 Trigonometry

1.7 Trigonometry

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Question 96

An ecologist monitors the population of Arctic hares, HHH hundreds, on a remote island at various times ttt months after a monitoring project began. The recorded data follows a cyclic pattern based on seasonal availability of vegetation. The researcher models this population using the equation:

H(t)=6.52cos⁡(2π(t−2.5)10.5)+14.34 H(t) = 6.52 \cos\left( \frac{2\pi(t - 2.5)}{10.5} \right) + 14.34 H(t)=6.52cos(10.52π(t−2.5)​)+14.34
a.

Find the minimum population of hares predicted by the model. Give your answer to the nearest whole hare.

[3]
b.

Find the duration of the longest continuous period during the first 21 months of the project (from t=0t=0t=0 to t=21t=21t=21) where the population predicted by the model is above 18 hundred. Give your answer in months to one decimal place.

[4]
c.

A different colony of hares in a more sheltered valley is found to have a more stable population with less variation between peaks and troughs, though it maintains the same average population and peak times. A student suggests refining the model for this sheltered colony by increasing the value 6.52 to 7.80. Explain whether this refinement is appropriate.

[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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