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ch(a ± b) = ch a ch b ± sh a sh b;
sh(a ± b) = sh a ch b ± ch a sh b;
th(a ± b) =
th a ± th b
1 ± th a th b
;
cth(a ± b) =
cth a cth b ± 1
cth b ± cth a
.
ch 2a = ch
2
a + sh
2
a; sh 2a = 2 sh a ch a;
th 2a =
2 th a
1 + th
2
a
; cth 2a =
1 + cth
2
a
2 cth a
.
ch a + ch b = 2 ch
a + b
2
ch
a − b
2
;
ch a − ch b = 2 sh
a + b
2
sh
a − b
2
;
sh a + sh b = 2 sh
a + b
2
ch
a − b
2
;
sh a − sh b = 2 sh
a − b
2
ch
a + b
2
;
th a ± th b =
sh(a ± b)
ch a ch b
; cth a ± cth b =
sh(b ∓ a)
sh a sh b
.
ch a ch b =
1
2
h
ch(a + b) + ch(a − b)
i
;
sh a sh b =
1
2
h
ch(a + b) − ch(a − b)
i
;
sh a ch b =
1
2
h
sh(a + b) + sh(a − b)
i
.
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3. Ôîðìóëû ñëîæåíèÿ àðãóìåíòîâ.
ch(a ± b) = ch a ch b ± sh a sh b;
sh(a ± b) = sh a ch b ± ch a sh b;
th a ± th b
th(a ± b) = ;
1 ± th a th b
cth a cth b ± 1
cth(a ± b) = .
cth b ± cth a
4. Ôîðìóëû äâîéíîãî àðãóìåíòà.
ch 2a = ch2 a + sh2 a; sh 2a = 2 sh a ch a;
2 th a 1 + cth2 a
th 2a = ; cth 2a = .
1 + th2 a 2 cth a
5. Ôîðìóëû ñëîæåíèÿ îäíîèìåííûõ ãèïåðáîëè÷åñêèõ ôóíê-
öèé.
a+b a−b
ch a + ch b = 2 ch ch ;
2 2
a+b a−b
ch a − ch b = 2 sh sh ;
2 2
a+b a−b
sh a + sh b = 2 sh ch ;
2 2
a−b a+b
sh a − sh b = 2 sh ch ;
2 2
sh(a ± b) sh(b ∓ a)
th a ± th b = ; cth a ± cth b = .
ch a ch b sh a sh b
6. Ôîðìóëû ïðåîáðàçîâàíèÿ ïðîèçâåäåíèÿ â ñóììó.
1h i
ch a ch b = ch(a + b) + ch(a − b) ;
2
1h i
sh a sh b = ch(a + b) − ch(a − b) ;
2
1h i
sh a ch b = sh(a + b) + sh(a − b) .
2
