| Dla dowolnych α, β ∈ R: sin(α+β)=sinαcosβ+cosαsinβ sin(α−β)=sinαcosβ−cosαsinβ cos(α+β)=cosαcosβ−sinαsinβ cos(α−β)=cosαcosβ+sinαsinβ Ponadto, dla tych kątów α, β, dla których dane wyrażenie jest określone, mamy: tg(α+β)=1−tg α⋅tg βtg α+tg β tg(α−β)=1+tg α⋅tg βtg α−tg β |
a)
sin975∘=sin(1080∘−105∘)=sin(3⋅360∘−105∘)=sin(−105∘)=−sin105∘=−sin(60∘+45∘)=−(sin60∘cos45∘+cos60∘sin45∘)=−(23⋅22+21⋅22)=−(46+42)=−46+2
sin(1231π)=sin(2127π)=sin(2π+127π)=sin(127π)=sin(123π+124π)=sin(4π+3π)=sin(4π)⋅cos(3π)+cos(4π)⋅sin(3π)=22⋅21+22⋅23=42+46=42+6
sin(−10125π)=sin(−10π−125π)=sin(−5⋅2π−125π)=sin(−125π)=−sin(125π)=−sin(122π+123π)=−sin(6π+4π)=−[sin(6π)⋅cos(4π)+cos(6π)⋅sin(4π)]=−(21⋅22+23⋅22)=−(42+46)=−42+6=4−2−6
b)
cos315∘=cos(360∘−45∘)=cos(−45∘)=cos45∘=22
cos(−495∘)=cos(−360∘−135∘)=cos(−135∘)=cos135∘=cos(90∘+45∘)=cos90∘⋅cos45∘−sin90∘⋅sin45∘=0⋅22−1⋅22=−22
cos(6125π)=cos(6π+125π)=cos(3⋅2π+125π)=cos(125π)=cos(122π+123π)=cos(6π+4π)=cos(6π)⋅cos(4π)−sin(6π)⋅sin(4π)=21⋅22−23⋅22=42−46=42−6
c)
tg 165∘=tg(180∘−15∘)=−tg 15∘=−tg(45∘−30∘)=−1+tg 45∘⋅tg 30∘tg 45∘−tg 30∘=−1+1⋅331−33=−1+3333−3=−33+333−3=−33−3⋅3+33=−3+33−3=−3+33−3⋅3−33−3=−9−3(3−3)2=−69−63+3=−612−63=−(2−3)=3−2
tg(−1005∘)=tg(−1080∘+75∘)=tg(−6⋅180∘+75∘)=tg 75∘=tg(45∘+30∘)=1−tg 45∘⋅tg 30∘tg 45∘+tg 30∘=1−1⋅331+33=33−333+3=33+3⋅3−33=3−33+3=3−33+3⋅3+33+3=9−3(3+3)2=69+63+3=612+63=2+3
tg(1213π)=tg(π+121π)=tg(121π)=tg(124π−123π)=tg(3π−4π)=1+tg(3π)⋅tg(4π)tg(3π)−tg(4π)=1+3⋅13−1=3+13−1=3+13−1⋅3−13−1=3−1(3−1)2=23−23+1=24−23=2−3
