General solutions of trigonometric equations
The line y = b meets the graph of sine, cosine or tangent in infinitely many points. Write all solutions as a formula with n ∈ ℤ, and in an equation that reduces to a quadratic reject the root that cannot occur.
- Ukraine: grade 10
- Russia: grade 10
Guide
In the Ukrainian programme (Grade 10, algebra and analysis, § 2, items 15-17) cos x = b is solved from the graph: the line y = b meets the cosine curve when |b| ≤ 1. The x-axis shows ordinary numbers, so π ≈ 3.14.
For cos x = b with |b| ≤ 1 the roots are x = ±arccos b + 2πn,n∈ℤ, where arccos b lies in [0;π] and arccos (−b) = π − arccos b. The special cases cos x = 0, cos x = 1, cos x = −1 are written without ±.
For sin x = b: x = (−1)narcsin b + πn,n∈ℤ. For tg x = b: x = arctg b + πn,n∈ℤ; for ctg x = b: x = arcctgb + πn,n∈ℤ.
When an equation reduces to a quadratic in sin x or cos x, the substitution t = sin x gives values of t; each must lie in [ − 1;1], otherwise there are no solutions.