Goniometrische vergelijkingen
• basisvergelijkingen
• sin(𝑥𝑥) = 𝑎𝑎 ⇔ 𝑥𝑥 = 𝛼𝛼 + 𝑘𝑘2𝜋𝜋 ⋁ 𝑥𝑥 = 𝜋𝜋 − 𝛼𝛼 + 𝑘𝑘2𝜋𝜋 met k ∈ ℤ, sin α = a, a ∈ [-1, 1]
• cos(𝑥𝑥) = 𝑎𝑎 ⇔ 𝑥𝑥 = 𝛼𝛼 + 𝑘𝑘2𝜋𝜋 ⋁ 𝑥𝑥 = −𝛼𝛼 + 𝑘𝑘2𝜋𝜋 met k ∈ ℤ, cos α = a, a ∈ [-1, 1]
• tan(𝑥𝑥) = 𝑎𝑎 ⇔ 𝑥𝑥 = 𝛼𝛼 + 𝑘𝑘𝑘𝑘 met k ∈ ℤ, tan α = a
• oplossingstechnieken
• uiteenvallen in basisvergelijkingen: sin 2𝑥𝑥 + cos 𝑥𝑥 = 0 ⇔ cos 𝑥𝑥 ∗ (2 sin 𝑥𝑥 + 1) = 0
• gebruik v. eenvoudige subsituties: 2 cos 2 𝑥𝑥 − cos 𝑥𝑥 + 1 = 0 ⇔ 2𝑡𝑡 2 − 𝑡𝑡 + 1 = 0
𝑐𝑐 𝑏𝑏
• vglk v/d vorm 𝑎𝑎 ∗ sin 𝑥𝑥 + 𝑏𝑏 ∗ cos 𝑥𝑥 = 𝑐𝑐: sin(𝑥𝑥 + 𝛼𝛼) = cos 𝛼𝛼 met tan 𝛼𝛼 =
𝑎𝑎 𝑎𝑎
• homogene vgl’en: afzonderen gem. factoren; delen door cos n 𝑥𝑥 met 𝑛𝑛 = graad v. overblijvende functie
𝑡𝑡 2 −1
• symmetische vgl’en (cos met sin wisselen: zelfde f): sin 𝛼𝛼 + cos 𝛼𝛼 = 𝑡𝑡, sin 𝛼𝛼 ∗ cos 𝛼𝛼 =
2
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