A sound engineer is analyzing a signal processing unit where the phase shift ϕ\phiϕ (in radians) depends on the frequency deviation fff. The relationship is given by the function:
ϕ(f)=arctan(f)+π2,f∈R \phi(f) = \arctan(f) + \frac{\pi}{2}, \quad f \in \mathbb{R} ϕ(f)=arctan(f)+2π,f∈RIdentify the graph of ϕ(f)\phi(f)ϕ(f) from the descriptions provided below.
Graph A: An increasing curve with horizontal asymptotes at ϕ=0\phi = 0ϕ=0 and ϕ=π\phi = \piϕ=π, passing through the point (0,π2)(0, \frac{\pi}{2})(0,2π).
Graph B: An increasing curve with horizontal asymptotes at ϕ=−π2\phi = -\frac{\pi}{2}ϕ=−2π and ϕ=π2\phi = \frac{\pi}{2}ϕ=2π, passing through the origin (0,0)(0, 0)(0,0).
Graph C: A decreasing curve with horizontal asymptotes at ϕ=π\phi = \piϕ=π and ϕ=0\phi = 0ϕ=0, passing through the point (0,π2)(0, \frac{\pi}{2})(0,2π).
Graph D: A curve restricted to the domain f∈[−1,1]f \in [-1, 1]f∈[−1,1] with endpoints at (−1,0)(-1, 0)(−1,0) and (1,π)(1, \pi)(1,π).
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.