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 (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.