A high-speed laser scanner rotates on a pivot 8 cm from a straight detector rail. The displacement h h\,h cm of the laser spot from the center of the rail is modelled by the function h(t)=8tan(πt15)\displaystyle h(t) = 8 \tan \left( \frac{\pi t}{15} \right)h(t)=8tan(15πt), where t t\,t is the time in microseconds. A digital cursor moves along the same rail with displacement z z\,z cm.
State the period of h(t)h(t)h(t).
Write down the number of points at which the laser spot and the cursor coincide (the number of roots of h(t)=z(t)h(t) = z(t)h(t)=z(t)) in the following cases:
(i) z(t)=8πt3\displaystyle z(t) = \frac{8\pi t}{3}z(t)=38πt in the interval −15<t<15-15 < t < 15−15<t<15
(ii) z(t)=4πt3\displaystyle z(t) = \frac{4\pi t}{3}z(t)=34πt in the interval −60<t<60-60 < t < 60−60<t<60
(iii) z(t)=4πt3\displaystyle z(t) = \frac{4\pi t}{3}z(t)=34πt in the interval −1500<t<1500-1500 < t < 1500−1500<t<1500
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.