Թեմա՝ երկու անկյունների գումարի և տարբերության եռանկյունաչափական ֆունկցիաների բանաձևեր, դասագիրք, առաջադրանքներ՝ 126-132
sin(α+β)=sin α*cos β + sin β*cos α
sin(α-β)=sin α*cos β — sin β*cos α
cos(α+β)=cos α*cos β-sin α*sin β
cos(α-β)=cos α*cos β+sin α*sin β
tg(α-β)=tg α — tg β/1+tg α*tg β
ctg(α-β)=ctg α*ctg β+1/ctg β-ctg α
Առաջադրանք 126
ա) cos(π/4-α)=cos π/4*cos α + sin π/4*sin α=√2/2*cos α + √2/2*sin α=√2/2*(cos α + sin α)
բ) cos(π/4+α)=cos π/4*cos α — sin π/4*sin α=√2/2*cos α — √2/2*sin α=√2/2*(cos α — sin α)
գ) sin(π/4-α)=sin π/4*cos α — sin α*cos π/4=√2/2*cos α — sin a*√2/2=√2/2*(cos α — sin α)
դ) sin(π/4+α)=sin*π/4*cos α + sin α * cos*π/4=√2/2*cos α + sin α*√2/2=√2/2*(cos α + sin α)
ե) tg(π/4+α)=tg 450+tg α/1-tg 450*tg α=1+tg α/1-1*tg α=(1+tg α)*(1-tg α)
զ) tg(π/4-α)=tg 450-tg α/1+tg 450*tg α=1-tg α/1+1*tg α=(1-tg α)*(1+tg α)
Առաջադրանք 127
ա) α=150 (450-300)
sin 450-300=sin A*cos B — cos A*sin B=sin450*cos300 — cos450*sin300=√2/2*√3/2 — √2/2*1/2=√6-√4/2=0.2588
cos 450-300=cos A*cos B + sin A*sin B=√2/2*√3/2 + √2/2*1/2=√6+√4/2
tg 450-300=1-1/√3 / 1+1/√3=√3-1/√3+1=2-√3
ctg 450-300=1/tg 150=2+√3
բ) α=750 (1200-450)
sin 1200-450=√3/2*√2/2 + 1/2*√2/2=√2(√3+1)/4
cos 1200-450=√2(√3-1)/4
tg 1200-450=2+√3
ctg 1200-450=2-√3
գ) α=1050 (600+450)
sin 600+450=√3/2*1/√2 + √1/2*1/2=√2(√3+1)/4
cos 600+450= -√2(√3-1)/4
tg 600+450= -2-√3
ctg 600+450= √3-2
դ) α=1650 (1200+450)
sin 1200+450=√2(√3-1)/4
cos 1200+450= -√2(√3+1)/4
tg 1200+450=√3-2
ctg 1200+450= -2-√3
Առաջադրանք 129
ա) √2 sin(π/4+α)-sin α=√2(sin π/4*cos α + cos π/4*sin α)-sin α
√2(√2/2*cos α + √2/2*sin α)-sin α
√2(√2*cos α + √2*sin α)/2 — sin α
2cos α + 2sin α/2 — sin α
2*(cos α + sin α)/2 — sin α
cos α+sin α-sin α=cos α
բ) √2 cos(π/4 — α)-cos α
√2(cos π/4*cos α + sin π/4*sin α)-cos α
√2(√2/2*cos α + √2/2*sin α)-cos α
√2*(√2cos α/2 + √2sin α/2)-cos α
√2*√2cos α/2 + √2sin α/2 — cos α
2cos α+2sin α/2 — cos α
cos α+sin α-cos α=sin α
գ) 2sin(π/6+α)-cos α=cos α + √3sin α — cos α=√3sin α
դ) √2cos α — 2cos(π/4+α)=√2cos α — √2cos α + √2sin α=√2sin α
Առաջադրանք 130
ա) √2cos(3π/4+α)+cos α / √2cos(5π/4-α)+sin α
√2(cos3π/*cos α — sin3π/4*sin α)+cos α / √2(cos5π/4*cos α — sin5π/4*sin α)+sin α
√2(-√2/2*cos α — √2/2*sin α)+cos α / √2(-√2/2*cos α — √2/2*sin α)+sin α
√2(-√2cos α/2 — √2sin α/2)+cos α / √2(-√2cos α/2 — √2sin α/2)+sin α
— √2(√2cos α + √2sin α)/2 + cos α : (-√2(√2cos α+√2sin α)/2 + sin α
-cos α — sin α + cos α / (-cos α — sin α + sin α)= -sin α/(-cos α)=tg α
բ) sin(2π/3+α)+1/2sin α / sin(7π/6-α)+1/2cos α
sin(2π/3+α)+sin α/2 / sin(7π/6-α)+cos α/2
(2sin(2π/3+α)+sin α)/2 : (2sin(7π/6-α)+cos α)/2
2sin(2π/3+α)+sin α / 2sin(7π/6-α)+cos α
√3cos α-sin α+sin α : 2*(-cos α+√3sin α)/2+cos α
√3cos α/√3sin α=cos α/sin α=ctg α
Առաջադրանք 131
ա) sin 270*cos 30 + cos 270*sin 30
sin 270+30=sin 300=1/2
բ) cos 870*cos 270 + sin 870*sin 270
cos 870-270=cos 600=1/2
Առաջադրանք 132
ա) (sin π/15+cos π/10)2 + (cos π/15+sin π/10)2=3
sin (π/15)2+2sin π/15*cos π/10+cos (π/10)2+cos (π/15)2+2cos π/15*sin π/10+sin (π/10)2=3
1+2sin π/15*cos π/10+1+2*1/2*(sin π/6-sin(-π/30))=3
2+(1/2-sin π/30)+(1/2+sin π/30)=3
2+1/2-sin π/30+1/2+sin π/30=3
2+1/2+1/2=2+1=3 (ճիշտ է)
բ) (cos π/9-cos 2π/9)2 + (sin π/9+sin2π/9)2=1
cos (π/9)2 -2cos π/9*cos 2π/9+cos (2π/9)2+sin (π/9)2+2sin π/9*sin 2π/9+sin (2π/9)2=1
1-2cos π/9*cos 2π/9+1+2*1/2*(cos (-π/9)-cos π/3)=1
2-2*1/2*(cos (-π/9)+cos π/3)+(cos π/9-1/2)=1
2-cos π/9-1/2+cos π/9-1/2=1 (ճիշտ է)
Դավիթ Մուրադյան 