22.
cos(a+β+γ)=
=cos(a+β)⋅cosγ−
−sin(a+β)⋅sinγ=
=(cosa⋅cosβ−sina⋅sinβ)⋅
⋅cosγ−
−(sina⋅cosβ+cosa⋅sinβ)⋅
⋅sinγ;
cosa=cos(arccos45)=45;
cosβ=cos(arccos1213)=1213;
cosγ=cos(arccos35)=35;
sina=1−(45)2=1−1625=
=925=35;
sinβ=1−(1213)2=1−144169=
=25169=513;
sinγ=1−(35)2=1−925=
=1625=45.
(cosa⋅cosβ−sina⋅sinβ)⋅
⋅sinγ=
=(45⋅1213−35⋅513)⋅35−
−(35⋅1213+45⋅513)⋅45=
=(4865−1565)⋅35−(3665+2065)⋅
⋅45=3365⋅35−5665⋅45=
=99325−224325=−125325=−513.
ОтветОтвет: −513.
бб)sin(arcsin35+arcsin513+arcsin45)=
=sin(a+β+γ)
sin(a+β+γ)=sin(a+β)⋅
⋅cosγ+cos(a+β)⋅sinγ=
=(sina⋅cosβ+cosa⋅sinβ)⋅
⋅cosγ+
+(cosa⋅cosβ−sina⋅sinβ)⋅
sina=sin(arcsin35)=35;
sinβ=sin(arcsin513)=513;
sinγ=sin(arcsin45)=45;
cosa=1−(35)2=1−925=
=1625=45;
cosβ=1−(513)2=1−25169=
=144169=1213;
cosγ=1−(45)2=1−1625=
=925=35.
(sina⋅cosβ+cosa⋅sinβ)⋅
+(cosa⋅cosβ−sina⋅sinβ)⋅sinγ=
=(35⋅1213+45⋅513)⋅35+
+(45⋅1213−35⋅513)⋅45=
=(3665+2065)⋅35+(4865−1565)⋅
⋅45=5665⋅35+3365⋅45=168325+
+132325=300325=1213.
ОтветОтвет:1213.
вв) ctg(arctg13+arctg14+arctg29)=
=ctg(a+β+γ)
\ ctg(a+β+γ)=
=−1+ctg(a+β)⋅ctg γctg(a+β)+ctgγ=
=−1+−1+ctga⋅ctgβctga+ctgβ⋅ctgγ−1+ctga⋅ctgβctga+ctgβ+ctgγ;
ctg a=1 :13=3;
ctg β=1 :14=4;
ctg γ=1 :29=92;
−1+−1+ctga⋅ctgβctga+ctgβ⋅ctgγ−1+ctga⋅ctgβctga+ctgβ+ctgγ=
=−1+−1+3⋅43+4⋅92−1+3⋅43+4+92=
=−1+117⋅9211∖27+9∖72=−1∖14+991422+6314=
=8514 :8514=1.
ОтветОтвет:1.