Calculate angles of a triangle given its sides

D”PZi MwYZ e¨envwiK 1g cÎ
cixÿv bs 1.1 ZvwiLt
cixÿ‡bi bvgt Ges we›`yØ‡qi ms‡hvRK †iLvsk‡K Abycv‡Z AšÍwe©f³Kvix we›`yi ¯’vbvsK
wbY©q|
g~jZZ¡t hw` Gi Dci we›`ywUi Rb¨ nq †hb Kv‡Z©mxq
Z‡j I `ywU we›`y nq Ges -A‡ÿi mgvšÍivj
nq Z‡e we›`y †K Abycv‡Z fvM Ki‡e| Avevi
I we›`y `yBwUi ms‡hvM †iLvsk‡K we›`ywU
Abycv‡Z AšÍwe©f³ Ki‡j, we›`yi ¯’vbv¼
( )
cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi
(iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi|
Kvh©c×wZt
1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv
I AuvwK|
2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK
a‡i Ges we›`y `yBwU emvB|
awi we›`yØq I | miæ †cwÝj w`‡q ms‡hvM K‡i
‡iLvsk †jLwP‡Î Dc¯’vcb Kwi|
3. we›`y w`‡q -A‡ÿi mgvšÍivj †iLvi Dci
GKK I GKK a‡i I `yBwU we›`y wbB|
4. †hvM Kwi Ges †iLv AsKb Kwi hv †K we›`y‡Z †Q` K‡i|
5. we›`yi ¯’vbvsK wPwýZ Kwi|
dj msKjbt
we›`yi ¯’vbvsK
MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb
(30 Ni, 10 Ni)=
( )
djvdjt cÖ`Ë we›`yØ‡qi ms‡hvM †iLvsk‡K Abycv‡Z AšÍwef³Kvix we›`yi ¯’vbvsK
gšÍe¨: MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK|
A(3,5)
C(6,2)
B(7,1)
P Q G
XOX
/
Y
Y
/
P
A
B
C
Q
1

cixÿY bs 1.2 ZvwiLt
cixÿ‡bi bvgt wÎfz‡Ri kxl©we›`y Ges wÎfzRwU †ÿÎdj wbY©q|
g~jZË¡t wÎfz‡Ri kxl©Îq Ges n‡j, wÎfzRwUi †ÿÎdj
| | eM© GKK|
cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK
K¨vjKz‡jUi|
Kvh©c×wZt
1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK|
2. Dfq Aÿ eivei ÿz`ªZg
e‡M©i evû = GKK a‡i
Ges
we›`y wZbwU emvB|
miæ †cwÝj w`‡q
ms‡hvM K‡i wÎfzRwU A¼b
Kwi|
3. we›`y w`‡q -A‡ÿi
mgvšÍivj †iLv AvuwK|
4. I n‡Z Gi
Ici h_vµ‡g I j¤^ AvuwK|
wnmvet we›`yi ¯’vbv¼ ( Ni, Ni)=
we›`yi ¯’vbv¼ ( Ni, Ni)=
wÎfz‡Ri †ÿÎdj
MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb
= UªwcwRqvg Gi †ÿÎdj+wÎfzR
Gi †ÿÎdj wÎfzR Gi †ÿÎdj =
eM©GKK
| | |
|
eM©GKK
djvdjt wÎfz‡Ri †ÿÎdj eM©GKK
gšÍe¨t MÖvd †_‡K cÖvß gvb I MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK|
B(-9, 1)
C(-3, -1)
M(5, -1)
L(5, 1)
A(5, 6)
P
Q
XX/
Y/
Y
O
2

cixÿ‡bi bvgt mij‡iLvi †jLwPÎ AsKb|
g~jZË¡t I m¤^wjZ mKj †hvMvkÖqx mgxKiY GKwU mij‡iLv wb‡`©k K‡i hLb I
DfqB GKB mv‡_ k~b¨ bq|
K¨vjKz‡jUi|
Kvh©c×wZt
1. cÖ`Ë mgxKiY n‡Z cvB-
mgxKiYwU‡Z Gi K‡qKwU gvb wb‡q Gi Abyiƒc gvb †ei Kwi I wb‡Pi QKwU •Zix Kwi:
3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i I we›`y¸wj ¯’vcb
Kwi Ges miæ †cwÝj w`‡q ms‡hvM K‡i mij‡iLvi †jLwPÎ AsKb Kwi|
‡jLwP‡Îi •ewkó¨t
i) cÖ`Ë mij‡iLvi Xvj-†Q` AvK…wZ I e‡j †iLvwU
Aÿ‡K abvZ¥K w`‡K GKK `~‡i †Q` Ki‡e|
ii) e‡j †iLvwU -A‡ÿi abvZ¥K w`‡Ki mv‡_ m~ÿè‡KvY Drcbœ K‡i|
C(-7, -2)
B(-2, 1)
A(3, 4)
3
X/
X
Y
Y/
O

cixÿv bs 2 ZvwiLt
cixÿ‡bi bvgt mgxKi‡Yi (e„Ë) †jLwPÎ AsKb|
g~jZË¡t mgxKiY GKwU e„‡Ëi mgxKiY hvi †K‡›`ªi ¯’vbvsK Ges
e¨vmva© |
cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii)
mv‡qw›UwdK K¨vjKz‡jUi|
Kvh©c×wZt
1. e„Ë‡K e„‡Ëi mv‡_ Zzjbv K‡i cvB,
e„‡Ëi †K›`ª Ges e¨vmva© =
3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i e„‡Ëi †K›`ª we›`yi Ae¯’vb wPwýZ
Kwi|
4. we›`y‡K †K›`ª K‡i GKK e¨vmva© wb‡q ‡cwÝj K¤úvm Gi mvnv‡h¨ e„Ë AsKb Kwi|
‡j‡Li •ewkó¨t
i. †jLwU GKwU e„Ë|
ii. †jLwU Awew”Qbœ|
iii. e„ËwU g~jwe›`yMvgx|
iv) e„ËwU Aÿ‡K GKwU we›`y‡Z Ges Aÿ‡K GKwU we›`y‡Z †Q` K‡i|
4
C(-3, 4)
XX/
Y/
Y

cixÿ‡bi bvgt GKwU wÎfz‡Ri evû¸wj h_vµ‡g †m.wg., †m.wg. Ges †m.wg. n‡j H wÎfz‡Ri e„nËg I
ÿz`ªZg †KvY wbY©q|
g~jZË¡t g‡b Kwi GKwU wÎfzR hvi wZbwU evû h_vµ‡g †m.wg., †m.wg. Ges
†m.wg. ‡Z e„nËg evû †m.wg. Gi wecixZ †KvY e„nËg †KvY Ges ÿz`ªZg evû
†m.wg. Gi wecixZ †KvY ÿz`ªZg †KvY| Zvn‡j cÖ`Ë Dcv‡Ëi mvnv‡h¨ AsKb K‡i Puv`vi mvnv‡h¨
e„nËg I ÿz`ªZg †Kvb wbb©q Kwi Ges m~Î I †_‡K cÖvß gv‡bi mv‡_
mZ¨Zv hvPvB Kwi|
cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj
K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi|
Kvh©c×wZt
1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I A¼b Kwi|
2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j awi|
3. MÖvd †ccv‡i eivei ÿz`ªZg e‡M©i evûi mgvb K‡i e„nËg evû ‡m.wg
†K‡U †bB|
4. ‡K †K›`ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q GKwU e„ËPvc AvuwK Ges †K
†K›`ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q AviI GKwU e„ËPvc AvuwK| e„ËPvcØq
ci¯úi we›`y‡Z †Q` K‡i| Ges †hvM Kwi| Zvn‡j †Z
‡m.wg., ‡m.wg Ges ‡m.wg m~wPZ K‡i|
5. Puv`vi mvnv‡h¨ e„nËg †KvY I ÿz`ªZg †KvY wbY©q Kwi|
wnmvet
dj msKjbt
e„nËg †KvY wbY©q ÿz`ªZg †KvY wbY©q
†m.wg. †m.wg. †m.wg.
MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb
djvdjt wb‡Y©q e„nËg †KvY Ges ÿz`ªZg †KvY
gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK|
5
XX/
Y/
Y
A
B
C

cixÿ‡Yi bvgt GKwU wÎfz‡Ri †KvY¸wj n‡j wÎfzRwUi evû¸wji AbycvZ wbY©q|
g~jZË¡t g‡b Kwi, Gi †KvY¸wj I Gi wecixZ evû¸wj
h_vµ‡g | Zvn‡j cÖ`Ë DcvË n‡Z MÖv‡di mvnv‡h¨ Ges m~‡Îi mvnv‡h¨ I
Gi AbycvZ wbY©q Kwi|
Kvh©c×wZt
1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK|
2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = †m.wg. †¯‹j awi|
3. MÖvd †ccv‡i eivei ‡m.wg †K‡U †bB|
4. Puv`vi mvnv‡h¨ we›`y‡Z I we›`y‡Z AsKb Kwi| I
‡iLv ci¯úi‡K we›`y‡Z †Q` K‡i|
5. MÖvd †_‡K Puv`vi mvnv‡h¨ Ges †cwÝj K¤úv‡mi mvnv‡h¨ I evûi •`N©¨ †g‡c eivei ewm‡q
h_vµ‡g I evûØ‡qi •`N©¨ wbY©q Kwi|
wnmvet Avgiv Rvwb, -‡Z
Avevi
dj msKjbt
djvdjt wb‡Y©q AbycvZ
gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK|
wbY©q
MÖvd †_‡K cÖvß AbycvZ M~Î †_‡K cÖvß AbycvZ
6
X/
X
Y
Y/

cixÿ‡Yi bvgt dvsk‡bi Ges iƒcvšÍwiZ ‡jLwPÎ AsKb K‡i †j‡Li •ewkó¨
wbY©q|
g~jZË¡t GKwU cive„‡Ëi mgxKiY hvi kxl©we›`y g~jwe›`y‡Z Ges Aÿ -Aÿ| Gi †jL
wb‡Ri mgvšÍiv‡j GKK ev‡g mwi‡q w`‡q cive„‡Ëi †jL cvIqv hvq hvi kxl©we›`y |
Kvh©c×wZt
2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit
3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges
miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi|
4. †jLwUi cÖwZwU we›`y‡K e‡M©i evûi mgvb A_©vr GKK evg w`‡K mwi‡q Gi
‡jL AsKb Kwi|
‰ewkó¨t
i. †jLwPÎ `yBwU cive„Ë| Gi kxl©we›`y Gi kxl© we›`y
ii. Gi †jL A‡ÿi mv‡c‡ÿ, I Gi †jL †iLvi mv‡c‡ÿ cÖwZmg|
iii. Dfq eµ‡iLvB wb‡Pi w`‡K †Lvjv  AvK…wZi cive„Ë|
7
(-3, -9)
(-1, -1)
0, 0
(1, -1)
(2, -4)
(3, -9)(-6, -9)
(-5, -4)
(-4, -1)
*-3, 0)X/ X
Y
Y/
(2,-1)
(-2,-4)
(-4,-1)
(0,-9)

cixÿ‡Yi bvgt GKB †jLwPÎ dvsk‡bi I Zvi wecixZ dvsk‡bi †jLwPÎ AsKb I †j‡Li
•ewkó¨ wbY©q|
g~jZË¡t †j‡Li Dci¯’ we›`y¸wji fzR I †KvwUi ¯’vb wewbgq K‡i Gi
†jLwPÎ AsKb Kiv hvq A_ev, ‡iLvi mv‡c‡ÿ Gi cÖwZ”Qwe AsKb K‡i
Gi ‡jL cvIqv hvq|
Kvh©c×wZt
-6
-7
miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj †hvM K‡i Gi †jL AsKb Kwi|
4. GB †¯‹‡j we›`y¸wj ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj †hvM K‡i
Gi †jL AsKb Kwi|
‣ewkó¨t
i. †jLwPÎ `yBwU mij †iLv wb‡`©k K‡i|
ii. ‡jLwPÎØq GKwU we›`y‡Z †Q` K‡i|
iii. ‡jLwPÎØq †iLvi mv‡c‡ÿ ci¯ú‡ii cÖwZ”Qwe|
8
(-6, -7)
(0, 5)
(1, 7)
(2, 9)
(-7, -6)
(5, 0)
(7, 1)
(9, 2)
X/
X
Y
Y/
(0,0)

cixÿY bs ZvwiLt
cixÿ‡Yi bvgt wÎ‡KvYwgwZK dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q hLb
g~jZË¡t myZivs GKwU ch©vqx
(Periodic) dvskb hvi ch©vq , ZvB cÖwZ e¨ewa‡Z Gi †jLwPÎ AweKj GKiƒc n‡e|
Kvh©c×wZ:
1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK|
3. -Aÿ eivei ÿz`ªZg e‡M©i evû = Ges -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i
ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i
Gi †jL AsKb Kwi|
‣ewkó¨t
i. †jLwPÎ Awew”Qbœ|
ii. Bnv †XD‡qi gZ|
iii. Bnv g~jwe›`yMvgx
iv. GKwU ch©vqe„Ë dvskb hvi ch©vq
9
X/
X
Y
Y/
O(0,0)
(900
,1)
(-1800
,0)
(-900
,1)
(1800
,0)

cixÿY bs ZvwiLt
cixÿ‡Yi bvgt ciggvb dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q|
g~jZË¡t mgxKi‡Y Gi mKj ev¯Íe gv‡bi Rb¨ Gi gvb AFbvZ¥K|
, hLb
= , hLb
K¨vjKz‡jUi|
Kvh©c×wZ:
miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í †hvM K‡i Gi †jL AsKb Kwi|
‣ewkó¨t
i. †jLwPÎwU †iLvi mv‡c‡ÿ cÖwZmg
ii. ‡jLwPÎwU 1g PZzf©vM I 2q PZzf©v‡M Amxg
ch©šÍ we¯Í…Z|
iii. †jLwPÎwU g~jwe›`y‡Z †Q` K‡i bv|
iv. †jLwPÎwU A‡ÿi abvZ¥K w`‡K we`¨gvb|
10
(-2, 5)
(-1, 3)
(0, 1)
(0.5, 0)
(1, 1)
(2, 3)
(3, 5)
X/
X
Y
Y/
O

cixÿY bs ZvwiLt
cixÿ‡Yi bvgt we›`yi mwbœK‡U dvsk‡bi †jL‡K Avmbœfv‡e H we›`y‡Z ¯úk©‡Ki †jL
Øviv ¯’vbxqfv‡e cÖwZ¯’vcb|
g~jZË¡t we›`yi mwbœK‡U dvskmb‡K Avmbœfv‡e H we›`y‡Z ¯úk©K Øviv ¯’vbwqfv‡e cÖwZ¯’vcb Kivi
m~Î,
K¨vjKz‡jUi|
Kvh©c×wZt
3. -Aÿ eivei ÿz`ªZg N‡ii evû = Ges -Aÿ eivei ÿz`ªZg N‡ii evû = GKK †¯‹j a‡i
ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i
Gi †jL AsKb Kwi|
4. we›`y‡Z ¯úk©K AsKb Kwi|
wnmvet
n‡Z cvB,
djvdjt we›`yi mwbœK‡U
dvsk‡bi
†jL‡K Avmbœfv‡e H we›`y‡Z ¯úk©K
Gi †jL Øviv ¯’vbxqfv‡e
cÖwZ¯’vcb Kiv nj|
11
(-90, -1)
(-60, -0.87)
(-30, -0.5)
(30, 0.5)
(60, 0.87)
(90, 1)
X/
X
Y
Y/
O(0,0)

cixÿY bs ZvwiLt
cixÿ‡Yi bvgt dvsk‡bi Rb¨ we›`y‡Z I wbY©q, ‡hLv‡b Ges †jLwP‡Î
I cÖ`k©b|
g~jZË¡t ¯^vaxb PjK I Aaxb Pj‡Ki AšÍi‡Ki ga¨Kvi m¤úK© Ges ¯^vaxb Pj‡Ki AwZÿz`ª
cwieZ©b Gi Rb¨ Aaxb Pj‡Ki AwZ ÿz`ª cwieZ©b
K¨vjKz‡jUi|
Kvh©c×wZt
3. -Aÿ I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi
Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi|
4. we›`y‡Z ¯úk©K AsKb Kwi| mij‡iLv‡K ¯úk©KwU Ges dvskbwU h_vµ‡g I we›`y‡Z
†Q` K‡i|
wnmvet
n‡Z cvB,
myZivs
Ges
wPÎ n‡Z cvB,
djvdjt I ‡jLwP‡Î cÖ`k©b Kiv n‡jv|
12
X/
X
Y
Y/
O(0,0)
A(2,4)
N
P
Q(3,9)
dy
y

cixÿY bs ZvwiLt
cixÿ‡Yi bvgt QqwU †KvwU eënvi K‡i ∫ Gi gvb wbY©q|
g~jZË¡t g‡b Kwi †ÿÎdj ∫
QqwU †KvwUi Rb¨ eënvi K‡i ∫ Gi gvb wbY©q Kwi|
cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi|
Kvh©c×wZt
1. eëwa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB eëwai wb¤œcÖvšÍ I
DaŸ©cÖv‡šÍi we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz`ª As‡ki †`N©¨ Gi gvb wbY©q Kwi|
2. Gi gvb n‡Z m~Î eënvi K‡i wbY©q Kwi †hLv‡b
3. ‡_‡K wbY©q Kwi|
4. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i
ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we›`y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi|
5. cÖvß QqwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi|
wnmvet
( )
= ( )
= eM© GKK (cÖvq)
djvdj:
wb‡Y©q †ÿÎdj ∫
= eM©GKK cÖvq|
gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ
ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e|
13
(0, 1) (0.16, 1.03)
(0.32, 1.11)
(0.48, 1.26)
(0.64, 1.51)
(0.8 1.9)
X
Y
Y
O
y0 y1 y2 y3 y4 y5
x0 x1 x2 x3 x4 x5

cixÿY bs ZvwiLt
cixÿ‡Yi bvgt cuvPwU †KvwU eënvi K‡i ∫ Gi gvb wbY©q|
g~jZË¡t g‡b Kwi, †ÿÎdj ∫
cuvPwU †KvwUi Rb¨ eënvi K‡i Gi gvb wbY©q Kwi|
K¨vjKz‡jUi|
Kvh©c×wZt
1. eëwa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB eëwai wb¤œcÖvšÍ I DaŸ© cÖv‡šÍi
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5. cÖvß cuvPwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi|
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= eM© GKK (cÖvq)
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wb‡Y©q †ÿÎdj ∫
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ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e|
14
XO
y0
x0
y1
x1
y2
x2
y3
x3
y4
x4
Y
(0,1)
(.25,.9412)
(.50,.80)
(.75,.64)
(1,.50)

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