An acoustic engineer is modelling the resonant response, R(θ)R(\theta)R(θ), of a signal filter where θ\thetaθ represents the phase angle in degrees.
Show that the response function
R(θ)=12sin2θsecθ+2cos2θcscθ R(\theta) = \frac{1}{2}\sin 2\theta \sec \theta + 2\cos 2\theta \csc \theta R(θ)=21sin2θsecθ+2cos2θcscθcan be simplified to the form
R(θ)=2cscθ−3sinθ R(\theta) = 2\csc \theta - 3\sin \theta R(θ)=2cscθ−3sinθwhere sinθ≠0\sin \theta \neq 0sinθ=0 and cosθ≠0\cos \theta \neq 0cosθ=0.
The engineer needs to find the phase angles where the response is exactly 5. A student attempts to solve the equation
12sin2θsecθ+2cos2θcscθ=5 \frac{1}{2}\sin 2\theta \sec \theta + 2\cos 2\theta \csc \theta = 5 21sin2θsecθ+2cos2θcscθ=5for 0∘≤θ≤360∘0^{\circ} \leq \theta \leq 360^{\circ}0∘≤θ≤360∘. They use the result from part (a) to produce the following steps:
Step 1: 2cscθ−3sinθ=52\csc \theta - 3\sin \theta = 52cscθ−3sinθ=5
Step 2: 2sinθ−3sinθ=5\frac{2}{\sin \theta} - 3\sin \theta = 5sinθ2−3sinθ=5
Step 3: 3sin2θ+5sinθ−2=03\sin^{2} \theta + 5\sin \theta - 2 = 03sin2θ+5sinθ−2=0
Step 4: sinθ=13\sin \theta = \frac{1}{3}sinθ=31 or sinθ=−2\sin \theta = -2sinθ=−2
Step 5: θ=19.5∘,160.5∘\theta = 19.5^{\circ}, 160.5^{\circ}θ=19.5∘,160.5∘ (to 1 decimal place)
Explain why the value sinθ=−2\sin \theta = -2sinθ=−2 must be rejected in Step 4.
Determine if there are any further reasons, based on the domain constraints of the original filter model, why specific solutions might need to be rejected, and state the final correct solutions for 0∘≤θ≤360∘0^{\circ} \leq \theta \leq 360^{\circ}0∘≤θ≤360∘.
317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.