Goniometrické úpravy
Jak upravím sinx + cosx = tg 150°
Odpovědi
Diskuze
sinx + cosx = √2*sin(x+π/4)
http://www.wolframalpha.com/input/?i=sinx+%2B+cosx+%3D
√2*sin(x+π/4) = tg 150°
x = arcsin( (tg 150°) / √2) - π/4
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Je to tg 135°
pak to bude:
x = arcsin( (tg 150°) / √2) - π/4
x = arcsin( -1 / √2) - π/4
x = arcsin( -√2 / 2 ) - π/4
sin -45°= ( -√2 / 2 )
π/4 ~ 180°/4 = 45°
x = arcsin( -√2 / 2) - π/4
x = - π/4 - π/4
x = - π/2 ~ -90°
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kontrola:
tg 135° = -1
sin(-π/2)+cos(-π/2)=-1
sin(-90°)+cos(-90°)=-1
( -1 ) + ( 0 )=-1
> OK
sinx + cosx = tg 135°
tg 135° = -1
sinx + cosx = -1
sinx + cosx = √2*sin(x+π/4)
===>
√2*sin(x+π/4)=-1
sin(x+π/4)=-√2/2
sin(?)=√2/2
sin(45°)=-sin(-45°)=√2/2
sin(π/4)=-sin(-π/4)=√2/2
sin(x+π/4)=-√2/2
sin(x+π/4)=sin(-π/4)
(a zde právě už nepotřebuji arcsin)
sin(x+π/4)=sin(-π/4)
(x+π/4)=(-π/4)
x=-π/4-π/4
x=-π/2
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x=-90°
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kontrola:
sin(-90°)+cos(-90°)=-1
OK ?
cos((π/2)-x) + cosx = -1