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.
- أوكرانيا: الصف ١٠
- روسيا: الصف ١٠
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الدليل
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| ≤ ١. The x-axis shows ordinary numbers, so π ≈ ٣٫١٤.
For cos x = b with |b| ≤ ١ the roots are x = ±arccos b + ٢πn,n∈ℤ, where arccos b lies in [٠;π] and arccos (−b) = π − arccos b. The special cases cos x = ٠, cos x = ١, cos x = −١ are written without ±.
For sin x = b: x = (−١)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 [ − ١;١], otherwise there are no solutions.