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 (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.