Dit document is een samenvatting van 'Module 2 - BEWIJS bijzondere hoeken', uit het boek 'NANDO 4D' voor het vak Wiskunde in het GO! Onderwijs in de doorstroomfinaliteit/ASO.
Bewijs
Gevraagd:
Goniometrische getallen van een hoek van 45°.
Oplossing:
Teken in de goniometrische cirkel een representant voor  = 45°.
ΔOPA is rechthoekig in P en gelijkbenig want de basishoeken Ô en  zijn gelijk aan 45°.
Dus: |OP| = |PA|
cos 45° = sin 45° (1)
We noteren de hoofdformule:
cos² 45° + sin² 45° = 1 -> (1)
2 cos² 45° = 1
1
cos² 45° =
2
1 √2
cos 45° = √2 = 2
(45° ∈ I, dus cos 45° > 0)
Besluit:
√2
sin 45° = cos 45° = 2
√2
sin 45°
tan 45° = = 2 =1
cos 45° √2
2
1
cot 45° = tan 45° = 1
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