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 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.