An optical sensor measures the angular deflection ϕ \phi\,ϕ of a light beam. During a calibration test, the system reaches equilibrium when the following relationship is satisfied:
7tanϕ=4cosϕ 7 \tan \phi = 4 \cos \phi 7tanϕ=4cosϕShow that this equilibrium condition can be expressed as the quadratic equation
4sin2ϕ+7sinϕ−4=0 4 \sin^2 \phi + 7 \sin \phi - 4 = 0 4sin2ϕ+7sinϕ−4=0A specific configuration of the sensor requires that the deflection varies such that ϕ=3ψ\phi = 3\psiϕ=3ψ. Determine all possible values of ψ \psi\,ψ in the range 0∘≤ψ≤60∘ 0^\circ \le \psi \le 60^\circ\,0∘≤ψ≤60∘ that satisfy the equilibrium condition:
7tan(3ψ)=4cos(3ψ) 7 \tan(3\psi) = 4 \cos(3\psi) 7tan(3ψ)=4cos(3ψ)Give your answers to 2 decimal places.
317 exam-style questions on AQA A Level Maths 1.8 E: Trigonometry, covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context. Each one has a worked solution and a mark scheme showing where the marks go.