Trigonometry
Trigonometry in NEB Class 12 tests inverse circular functions, general solutions of trigonometric equations, and properties of triangles, including the sine rule, cosine rule, projection and half-angle formulas. The exam expects both principal values and general families of solutions.
Core rules
- Principal ranges: sin⁻¹x ∈ [-π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (-π/2, π/2).
- Identities: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] when the denominator is nonzero.
- General solutions: if sin θ = sin α, then θ = nπ + (-1)^n α; if cos θ = cos α, then θ = 2nπ ± α; if tan θ = tan α, then θ = nπ + α, where n is an integer.
- Triangle: a/sin A = b/sin B = c/sin C = 2R; a² = b² + c² - 2bc cos A; area = (1/2)ab sin C.
Worked example
Solve 2 sin²x + 3 cos x = 0 for general x. Replace sin²x = 1 - cos²x: 2 - 2cos²x + 3cos x = 0, so 2cos²x - 3cos x - 2 = 0. Factor: (2cos x + 1)(cos x - 2) = 0. Since cos x = 2 is impossible, cos x = -1/2, giving x = 2nπ ± 2π/3.
Traps
- Giving only the principal value when the question asks for the general solution.
- Dropping the ± in cosine solutions or the (-1)^n in sine solutions.
- Mixing degrees and radians in the same computation.
- Squaring an equation and accepting extraneous roots without checking.
- Using the sine rule when two sides and the included angle are given; the cosine rule is then required.