A high-speed laser sensor rotates such that the horizontal displacement, d d\,d cm, of the laser spot on a wall is modeled by the function d=tan(3t)d = \tan(3t)d=tan(3t), where t t\,t is the time in seconds. A target moves along the same wall with a displacement modeled by the linear equation d=ktd = ktd=kt for various values of kkk.
State the period of the function tan(3t)\tan(3t)tan(3t).
Determine the number of roots of the equation:
(i) tan(3t)=60t\tan(3t) = 60ttan(3t)=60t in the interval −π3<t<π3\displaystyle -\frac{\pi}{3} < t < \frac{\pi}{3}−3π<t<3π
(ii) tan(3t)=30t\tan(3t) = 30ttan(3t)=30t in the interval −2π<t<2π-2\pi < t < 2\pi−2π<t<2π
(iii) tan(3t)=30t\tan(3t) = 30ttan(3t)=30t in the interval −40π<t<40π-40\pi < t < 40\pi−40π<t<40π
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.