Trigonometry and Complex Numbers
Trigonometry and complex numbers tests exact identities, general solutions, inverse trigonometric functions, and the geometric algebra of complex numbers. Scholarship questions often link polar form and roots to loci or trigonometric sums.
Core rules and concepts
- General solutions: sin θ = sin α gives θ = α + 2kπ or θ = π - α + 2kπ; cos θ = cos α gives θ = ±α + 2kπ; tan θ = tan α gives θ = α + kπ.
- Inverse trig functions have restricted ranges, e.g. arcsin: [-π/2,π/2], arccos: [0,π].
- Complex modulus r=|z|≥0 and argument arg z is multivalued; choose the principal value in (-π,π] or [0,2π) as specified.
- De Moivre: (r cis θ)^n = r^n cis(nθ); nth roots are r^{1/n} cis((θ+2kπ)/n), k=0,...,n-1.
- Loci: |z-a|=r is a circle; |z-a|=|z-b| is the perpendicular bisector.
Worked example
Solve z^3 = 8i. Write 8i = 8 cis(π/2). Then z = 2 cis((π/2+2kπ)/3), k=0,1,2. These are 2 cis(π/6)=√3+i, 2 cis(5π/6)=-√3+i, and 2 cis(3π/2)=-2i.
Traps
Using the wrong quadrant for arg; dividing only θ by n and forgetting 2kπ; squaring equations and accepting extraneous roots; writing cis θ = cos θ + i sin θ but dropping i; assuming arg(z1z2)=arg z1+arg z2 without modulo 2π.