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=4cosϕ7 \tan \phi = 4 \cos \phi7tanϕ=4cosϕ
Show that this equilibrium condition can be expressed as the quadratic equation 4sin2ϕ+7sinϕ−4=04 \sin^2 \phi + 7 \sin \phi - 4 = 04sin2ϕ+7sinϕ−4=0
A 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.
Practise Edexcel A Level Maths 7.3 Solving Trigonometric Equations with exam-style questions for A Level Maths. 29 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.