A high-precision sensor measures the output voltage V V\,V of a microscopic resonator as a function of its deflection angle θ\thetaθ (in radians). The sensor output is modeled by the equation:
V(θ)=3cos(2θ)−3cos(2θ)cos(6θ) V(\theta) = 3\cos(2\theta) - 3\cos(2\theta)\cos(6\theta) V(θ)=3cos(2θ)−3cos(2θ)cos(6θ)Show that for small values of θ\thetaθ, V(θ)≈54θ2V(\theta) \approx 54\theta^2V(θ)≈54θ2.
The energy E E\,E dissipated during an oscillation cycle is given by E=∫00.1154V(θ) dθ\displaystyle E = \int_{0}^{0.1} \sqrt{\frac{1}{54}V(\theta)} \, d\thetaE=∫00.1541V(θ)dθ. Show that the energy E E\,E can be approximated by E≈2m×5nE \approx 2^m \times 5^nE≈2m×5n, where m m\,m and n n\,n are integers to be determined.
Explain why ∫12.612.7θ dθ\int_{12.6}^{12.7} \theta \, d\theta∫12.612.7θdθ is not a suitable approximation for ∫12.612.7154V(θ) dθ\displaystyle \int_{12.6}^{12.7} \sqrt{\frac{1}{54}V(\theta)} \, d\theta∫12.612.7541V(θ)dθ.
Explain how ∫12.612.7154V(θ) dθ\displaystyle \int_{12.6}^{12.7} \sqrt{\frac{1}{54}V(\theta)} \, d\theta∫12.612.7541V(θ)dθ may be approximated by ∫abθ dθ\int_{a}^{b} \theta \, d\theta∫abθdθ for suitable values of a a\,a and bbb.
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.