Matematica per le applicazioni I - Trigonometria

Coordinate dei punti simmetrici rispetto all'origine O

Oπ + α′P′PP sono due punti simmetrici rispetto all'origine O, le cui coordinate sono:

(x, y) = (π - α, π + αα)

Per quanto detto in precedenza sulle simmetrie:

(π + α) = -αcos(θ)

(π + α) = -αsen(θ)

Pertanto, le coordinate di P saranno:

(x, y) = (-α, -α′P cos(θ), -α′P sen(θ))

Sostituendo alla tangente avremo:

(π + α) = αtg(θ)

Esempi:

Esercizio riguardante archi che differiscono di mezzo giro:

(α, π + α + α) = (α - α + α - α) = sin(θ) cos(θ) 1

α + α + =2

2sin cos 1= − + =1

1 0Esercizio riguardante archi che differiscono di mezzo giro:( )π + αcos1 +( ) ( )+ π + α − π + α21 cos 1 cos− α1 cos= + =− α − α21 cos 1 cos+ α − α1 1 1cos cos= = =− α − α α2 2 21 cos 1 cos sinEsercizio riguardante archi che differiscono di mezzo giro:( )α − π + αsin sin2 ( )α − π + αcos cos2 ( )α α− −sin sin2= =( )α α− −cos cos2α α α+ 3sin sin sin2 α= = = tgα α α+cos 2 cos 3 cosANGOLI ESPLEMENTARI ( )2πDue angoli si dicono esplementari quando la loro somma dà un angolo giroAppunti a cura di Roberto Bringheli e Carmelo Zucco easy matematica di Adolfo ScimonePagina 10 di 15Pπ − α2 α ′P′PP e sono due punti opposti (simmetrici rispetto all’asse

x) le cui coordinate sono:[ ]( ) ( )( ) π − α π − αα α ′P cos 2 ;sen 2P cos ;sene per quanto detto in precedenza sulle simmetrie avremo:( )α − α′P cos ; sensostituendo alla tangente avremo:( )π α− α−sen 2 sen( )π α α− = = = −tg 2 tg( )π α α−cos 2 cosdunque:( )π α α− = −tg 2 tgEsempi riguardanti angoli esplementari:Esercizio n° 1− − − + − −π α π α α αsin ( 2 ) 2 cos( 2 ) cos cos( )= − − + − =α α α αsin 2 cos cos cos= − −α αsin 2 cosEsercizio n° 2 2 ( )π α π α α− + + − −tg(2 ) 2sin(2 ) tgcπ α−cos(2 )2 ( )α α α= − + + − =tg 2sin tgcαcosα α sin 2 cosα=− + + −

= 2sinα α α cos cos sinαsin 2 α=− + − =1cosα αcos cosα α− + − 2sin 2 2 cos= =αcos( )α α− + − 2 α α− +sin 2 1 cos 2sin 2sin= =α αcos cos

Esercizio n° 3

Appunti a cura di Roberto Bringheli e Carmelo Zucco easy matematica di Adolfo Scimone

Pagina 11 di 15

π − α + π + αsin ( 2 ) cos( 2 )( )− α + π − αcos sin ( )− α − αsin cos= =α + αcos sin( )α α− +cos sin= = −1α α+cos sin

ARCHI OPPOSTI α − α

Due archi si dicono opposti quando i due angoli formati hanno ampiezza ey Pα− α xO ′P′PP e sono due punti (simmetrici rispetto all’asse x) le cui ordinate sono:( ) ( )α − αα α ′P cos ; senP cos ;sen

dunque per quanto detto in precedenza sulle

simmetrie avremo: (α) = αcos(θ) cos(α) - αsin(θ) sin(α) sostituendo alla tangente avremo: α = -tg(θ)tg(α) - cos(θ)cos(α) dunque: (α) = -tg(θ)tg(α) Esempi Esercizio riguardante angoli opposti: (α) = sin(θ)cos(α) - 2sin(θ)cos(3α) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ)cos(α) + sin(2α)cos(θ) (α) = sin(θ) cos= ⋅ =α +α α2 2cos sincos+α α α α1sin sin(cos ) cos= ⋅ =α 1cos= +α α2 1sin (cos )

ARCHI COMPLEMENTARI

Due archi si dicono complementari quando la somma dei due angoli formati è un angolo retto.

′P PO′PP e sono due punti(simmetrici rispetto alla bisettrice del 1° e 3° quadrante) le cui coordinate sono: π π    ( ) α α− −α α ′    P cos ;senP cos ;sen       2 2dunque per quanto detto in precedenza sulle simmetrie avremo:π  =− α α cos sen 2π − =α α sen sen 2sostituendo alla tangente avremo:Appunti a cura di Roberto Bringheli e Carmelo Zucco easy matematica di Adolfo ScimonePagina 13 di 15π α− senπ α    cos2α α− = = = tg c tgπ α

² senα−²cos²dunque:π^α α− =^αtg c tg²Esempi πEsercizio riguardante archi complementari, opposti e che differiscono di :π^α + − α + − α − π + α^{sin cos 3 cos( ) sin ( )}= α + α + α + α =sin sin 3 cos sin= α + α = α + α3sin 3 cos 3( sin cos ) πEsercizio riguardante archi complementari, opposti e che differiscono di :π π π^{( )− − + − −α α α π α} 2cos sin tg sin²= + ⋅ =α α α α2cossin ctg sinαcos= + ⋅ =α α α2cossin sinαsin= α α2 sin cos πEsercizio riguardante archi complementari, opposti e che differiscono di :π π + α π   sin ( )α α − + −

α−  Tg tg ctg   −π αcos( )2 2− αsin= ⋅ − + =α α αctg tg tg− αcosα α α αcos sin sin sin= ⋅ − + =α α α αcos cos cossin α αsin sin= − + =1 1α αcos cos

ARCHI CHE DIFFERISCONO DI UN ANGOLO RETTO

Appunti a cura di Roberto Bringheli e Carmelo Zucco easy matematica di Adolfo Scimone

Pagina 14 di 15

y′P Qπ + α2 Pα x′ ′P P

Per ottenere le coordinate di bisogna effettuare 2 passaggi, passare da P a Q e da Q a .

Q e P sono due punti (simmetrici rispetto alla bisettrice del 1° e 3° quadrante) le cui( ) ( )α α α αP cos ;sen Q sen ;cos per quanto detto in precedenzacoordinate sono

sulle simmetrie:

π π    + α + α′    P cos ;sen     2 2dunque per quanto detto in precedenza sulle

simmetrie avremo: _π ^ = −+ α α^{cos sen 2π + =α α sen cos 2sostituendo alla tangente avremo_π α+ sen_π α    cos2α α+ = = = − }