Wzory redukcyjne:
1) tg (180∘−α)=−tg α, jesˊli α∈⟨0∘, 90∘)∪(90∘, 180∘⟩
2) ctg (180∘−α)=−ctg α, jesˊli α∈(0∘, 180∘)
3) sin(180∘−α)=sinα, jesˊli α∈⟨0∘, 180∘⟩
4) cos(180∘−α)=−cosα, jesˊli α∈⟨0∘, 180∘⟩
5) tg (90∘+α)=−ctg α
6) ctg (90∘+α)=−tg α
7) sin(90∘+α)=cosα
8) cos(90∘+α)=−sinα
9) tg (90∘−α)=ctg α
10) ctg (90∘−α)=tg α
11) sin(90∘−α)=cosα
12) cos(90∘−α)=sinα
Wzory 5) - 12) zachodzą dla α∈(0∘, 90∘).
a) tg 43∘⋅tg 44∘⋅tg 45∘⋅tg 45∘⋅tg 47∘=tg 45∘=1
=tg 43∘⋅tg 44∘⋅tg 46∘⋅tg 47∘=
=tg (90∘−47∘)⋅tg (90∘−46∘)⋅tg 46∘⋅tg 47∘=9)
=ctg 47∘⋅= 1(ctg 46∘⋅tg 46∘)⋅tg 47∘=
=ctg 47∘⋅1⋅tg 47∘=ctg 47∘⋅tg 47∘=1
b) ctg 25∘⋅ctg 35∘⋅ctg 45∘⋅ctg 55∘⋅ctg 65∘=ctg 45∘=1
=ctg 25∘⋅ctg 35∘⋅ctg 55∘⋅ctg 65∘=
=ctg (90∘−65∘)⋅ctg (90∘−55∘)⋅ctg 55∘⋅ctg 65∘=10)
=tg 65∘⋅= 1(tg 55∘⋅ctg 55∘)⋅ctg 65∘=
=tg 65∘⋅ctg 65∘=1
c) sin275∘+sin215∘−2sin30∘=sin30∘=21
=sin275∘+sin215∘−2⋅21=sin275∘+sin215∘−1=
=sin2(90∘−15∘)+sin215∘−1=11)
== 1(cos215∘+sin215∘)−1=1−1=0
d) (cos52∘−cos38∘)2+2sin38∘⋅sin52∘+2cos60∘=cos60∘=21
=(cos52∘−cos38∘)2+2sin38∘⋅sin52∘+2⋅21=
=(cos52∘−cos38∘)2+2sin38∘⋅sin52∘+1=
=cos252∘−2cos52∘⋅cos38∘+cos238∘+2sin38∘⋅sin52∘+1=
=cos252∘−2cos52∘⋅cos38∘+cos238∘+2sin(90∘−52∘)⋅sin(90∘−38∘)+1=11)
=cos252∘−2cos52∘⋅cos38∘+cos238∘+2cos52∘⋅cos38∘+1=
=cos252∘+cos238∘+1=
=cos2(90∘−38∘)+cos238∘+1=12)
== 1(sin238∘+cos238∘)+1=1+1=2
