Trikonmiti Ke Prayog Class 10 Notes: Unchai Aur Doori Wale Sawal Ab Mushkil Nahi Lagenge
Class 10 ke Maths me Chapter 8 me aapne sin, cos aur tan seekhe the. Lekin Chapter 9 me asli khel shuru hota hai — jab ye ratios kisi tower ki unchai, kisi patang ki doori ya kisi jahaz ki sthiti nikalne ke kaam aate hain. Kai students ko trikonmiti ke prayog class 10 notes isliye mushkil lagte hain, kyunki unhe samajh nahi aata ki diagram kaise banayein aur kaunsa ratio kahan lagayein. Is article me wahi dikkat door hogi: pehle teen basic terms, phir step-by-step solved examples, aur ant me practice questions jinke answers bhi diye gaye hain.
Ek baat pehle hi saaf kar dein: ye notes NCERT Class 10 Maths ke Chapter 9 “Trikonmiti ke Kuch Prayog” (Some Applications of Trigonometry) par aadharit hain. Ye Chapter 8 ke “Trikonmiti ka Parichay” se alag hai, jiske notes site par pehle se maujood hain. Chapter 8 me aapne ratios seekhe the, Chapter 9 me un ratios ka asli jeevan me istemal seekhenge. Trikonmiti ke prayog class 10 notes ko samajhne ke liye Chapter 8 ke ratios yaad hona zaroori hai, isliye shuruat me unki ek chhoti revision bhi di gayi hai.
Poori Jaankari Ek Nazar Me
| Baat | Jaankari |
|---|---|
| Subject | Class 10 Mathematics |
| Chapter | Chapter 9 — Trikonmiti ke Kuch Prayog (Some Applications of Trigonometry) |
| NCERT book | Mathematics, Class 10 |
| Board | CBSE aur state boards (NCERT syllabus) |
| Isme kya cover hai | Unnayan kon, avnaman kon, unchai-doori ke solved examples, practice questions |
| Kin exams ke liye | Board Exam 2027, unit tests, school exams |
| Zaroori pehla gyaan | Chapter 8 ke ratios: sin, cos, tan aur standard angles (30°, 45°, 60°) |
Trikonmiti Ke Prayog Class 10 Notes: Teen Terms Jo Har Sawal Me Aayenge
Har heights-and-distances wale sawal me teen shabdon ka istemal hota hai, aur inhe samajhe bina diagram banana mushkil hai. Pehla shabd hai “drishti rekha” (line of sight) — ye wo seedhi rekha hai jo dekhne wale ki aankh se dekhi jaane wali vastu tak jaati hai. Doosra shabd hai “unnayan kon” (angle of elevation) — jab vastu dekhne wale ki aankh ke star se upar hoti hai, to drishti rekha aur kshitij rekha (horizontal line) ke beech jo kon banta hai, use unnayan kon kehte hain.
Teesra shabd hai “avnaman kon” (angle of depression) — jab vastu dekhne wale ki aankh ke star se neeche hoti hai, to drishti rekha aur kshitij rekha ke beech jo kon banta hai, use avnaman kon kehte hain. Ek bahut kaam ki baat yaad rakhein: avnaman kon hamesha us kon ke barabar hota hai jo neeche wali vastu se upar ki taraf banta hai, kyunki ye alternate interior angles hote hain. Isliye avnaman kon wale sawalon me aap seedha samkon tribhuj me wahi kon istemal kar sakte hain. Trikonmiti ke prayog class 10 notes ka ye sabse zyada istemal hone wala trick hai.
Revision: Standard Angles Ke Ratios
Chapter 9 ke har sawal me 30°, 45° aur 60° ke ratios kaam aate hain. Inhe ratne ki bajay samajh kar yaad rakhein:
| Kon | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
In ratios se teen formulas bante hain jo har sawal me kaam aate hain. Samkon tribhuj me tan θ = lamb / aadhar, sin θ = lamb / karn, aur cos θ = aadhar / karn. Heights-and-distances ke zyadatar sawalon me tan ka istemal hota hai, kyunki unchai (lamb) aur doori (aadhar) hi aksar diye ya poochhe jaate hain. Jab patang ki dori ya seedhi ki lambai (karn) di ho to sin ya cos ka istemal hota hai. Trikonmiti ke prayog class 10 notes me aapko ye pehchan karna seekhna hai ki sawal me kaunsi do bhujaen judi hui hain.
Trikonmiti Ke Prayog Class 10 Notes: Solved Example 1 — Tower Ki Doori
Sawal: Ek 30 metre unche tower ke shikhar ka unnayan kon zameen par ek bindu se 30° hai. Us bindu ki tower ke paad (foot) se doori kitni hai?
Hal: Pehle diagram banayein. Tower ko AB manein, jahan A tower ka paad hai aur B shikhar hai, isliye AB = 30 m. Zameen par bindu C hai, jahan se unnayan kon 30° hai. Tribhuj ABC me kon A = 90° hai, kyunki tower zameen par seedha khada hai. Hamein AC nikalna hai.
Tribhuj ABC me tan 30° = AB / AC. Tan 30° ka maan 1/√3 hai. Isliye 1/√3 = 30 / AC. Cross multiply karne par AC = 30√3 metre. √3 ka maan lagbhag 1.732 hota hai, isliye AC = 30 × 1.732 = 51.96 metre. Jawab: bindu ki tower ke paad se doori lagbhag 51.96 metre hai.
Check: 51.96 / 30 = 1.732 = √3, aur tan 30° = 1/√3 ka ulta √3 hi hota hai jab lamb aur aadhar ki jagah badal jaye. Hisaab sahi hai.
Solved Example 2 — 45° Wala Aasan Sawal
Sawal: Zameen par ek bindu se 20 metre door khadi imarat ke shikhar ka unnayan kon 45° hai. Imarat ki unchai kitni hai?
Hal: Imarat ko AB manein, AB = h (gyat karna hai). Bindu C, A se 20 metre door hai, isliye AC = 20 m. Tribhuj ABC me kon A = 90° aur kon C = 45° hai. Tan 45° = AB / AC = h / 20. Tan 45° ka maan 1 hota hai, isliye h / 20 = 1, yani h = 20 metre.
Ye sawal isliye mahatvapoorna hai, kyunki 45° ke case me unchai aur doori barabar ho jaati hain. Board exam me 45° wale sawal aksar aate hain aur ye sabse jaldi hal hote hain. Agar unnayan kon 45° dikhe to turant samajh jayein ki lamb aur aadhar barabar hain. Trikonmiti ke prayog class 10 notes me ye observation bahut samay bachati hai.
Solved Example 3 — Patang Ki Unchai
Sawal: Ek patang ki dori 100 metre lambi hai aur wo kshitij se 60° ka kon banati hai. Patang zameen se kitni unchai par hai? (Dori ko seedha mana jaye.)
Hal: Yahan dori karn ka kaam kar rahi hai, isliye sin ka istemal hoga. Maan lein patang ki unchai h hai. Samkon tribhuj me sin 60° = lamb / karn = h / 100. Sin 60° ka maan √3/2 hai. Isliye h / 100 = √3/2, yani h = 100 × √3/2 = 50√3 metre. √3 = 1.732 rakhne par h = 50 × 1.732 = 86.60 metre.
Check: 86.60 / 100 = 0.866, jo sin 60° ke barabar hai. Hisaab sahi hai. Is sawal se ye seekhne ko milta hai ki jab karn di ho aur lamb poochha gaya ho to sin lagta hai, aur jab karn di ho aur aadhar poochha gaya ho to cos lagta hai. Trikonmiti ke prayog class 10 notes me ratio chunne ki ye pehchan sabse zaroori skill hai.
Solved Example 4 — Avnaman Kon
Sawal: 30 metre unche tower ke shikhar se ek car ka avnaman kon 30° hai. Car ki tower ke paad se doori kitni hai?
Hal: Tower AB = 30 m, shikhar B se car C ka avnaman kon 30° hai. Alternate interior angles ke niyam se, car ke bindu par bana kon bhi 30° hoga, yani tribhuj ABC me kon C = 30° hai. Tan 30° = AB / AC = 30 / AC. Tan 30° = 1/√3, isliye AC = 30√3 = 30 × 1.732 = 51.96 metre.
Ye sawal batata hai ki avnaman kon ko dekh kar ghabrana nahi hai. Avnaman kon ko seedha neeche wale tribhuj me utaar dein, phir sawal bilkul unnayan kon jaisa hi ban jaata hai. Board exam me avnaman kon wale sawal aksar aate hain aur jo students is trick ko jaante hain, wo inhe sabse jaldi hal karte hain.
Solved Example 5 — Do Binduon Wala Sawal
Sawal: Ek khambe ke shikhar ka unnayan kon ek bindu se 60° hai. Us bindu se 20 metre door jaane par unnayan kon 30° ho jaata hai. Khambe ki unchai kitni hai?
Hal: Khambe ki unchai h manein. Pehle bindu ki khambe ke paad se doori x manein. Pehle tribhuj me tan 60° = h / x, yani √3 = h / x, isliye h = x√3. Doosre bindu par doori (x + 20) ho gayi aur kon 30° hai, isliye tan 30° = h / (x + 20), yani 1/√3 = h / (x + 20), isliye h = (x + 20) / √3.
Dono h barabar hain, isliye x√3 = (x + 20) / √3. Dono taraf √3 se guna karne par 3x = x + 20, yani 2x = 20, isliye x = 10 metre. Ab h = x√3 = 10√3 = 10 × 1.732 = 17.32 metre. Jawab: khambe ki unchai lagbhag 17.32 metre hai.
Check: doosre bindu se doori 30 metre hai, tan 30° = 17.32 / 30 = 0.577 = 1/√3. Hisaab sahi hai. Do binduon wale sawalon me hamesha do equations banayein aur unhe barabar rakhein — yehi standard tareeka hai.
Solved Example 6 — Lighthouse Aur Do Jahaz
Sawal: 40 metre unche lighthouse ke shikhar se do jahazon ke avnaman kon 60° aur 30° hain, aur dono jahaz lighthouse ke ek hi taraf hain. Dono jahazon ke beech ki doori kitni hai?
Hal: Lighthouse AB = 40 m. Pehla jahaz C, jiska avnaman kon 60° hai, aur doosra jahaz D, jiska avnaman kon 30° hai. Alternate angles se tribhuj ABC me kon C = 60° aur tribhuj ABD me kon D = 30° hai. Pehle jahaz ki doori: tan 60° = 40 / AC, yani √3 = 40 / AC, isliye AC = 40 / √3 = 40 × 1.732 / 3 = 23.09 metre. Doosre jahaz ki doori: tan 30° = 40 / AD, yani 1/√3 = 40 / AD, isliye AD = 40√3 = 69.28 metre. Dono jahazon ke beech ki doori = AD − AC = 69.28 − 23.09 = 46.19 metre.
Check: 40√3 − 40/√3 = 40 × (3 − 1) / √3 = 80 / √3 = 80 / 1.732 = 46.19. Hisaab sahi hai. Is sawal me ye dhyaan rakhna zaroori hai ki bada avnaman kon wali vastu nazdeek hoti hai aur chhota kon wali door hoti hai.
Aam Galtiyan
Pehli galti: unnayan aur avnaman kon ko ulta samajhna. Unnayan kon hamesha upar dekhne par banta hai aur avnaman kon neeche dekhne par. Doosri galti: diagram na banana. Bina diagram ke sawal hal karna mushkil hota hai, isliye pehle saaf diagram banayein aur di gayi values us par likhein.
Teesri galti: galat ratio chunna. Yaad rakhein: lamb aur aadhar ka sawal ho to tan, lamb aur karn ka ho to sin, aadhar aur karn ka ho to cos. Chauthi galti: √3 ka maan galat lena. √3 = 1.732 hota hai, 1.73 ya 1.732 dono board me maane jaate hain, lekin beech ke hisaab me √3 ko surd ke roop me hi rakhein aur antim jawab me hi decimal me badlein, warna rounding error aa sakta hai. Paanchvi galti: degree aur radian ko confuse karna — board ke sawal degree me hi hote hain.
Practice Questions (Answers Ke Saath)
1. Zameen par ek bindu se 50 metre door khadi imarat ke shikhar ka unnayan kon 45° hai. Imarat ki unchai kitni hai? Jawab: 50 metre (tan 45° = 1, isliye unchai = doori).
2. Ek patang ki dori 60 metre lambi hai aur wo kshitij se 60° ka kon banati hai. Patang ki unchai kitni hai? Jawab: 30√3 = 51.96 metre (sin 60° = h / 60).
3. 25 metre unchi chattan ke shikhar se ek naav ka avnaman kon 45° hai. Naav ki chattan ke paad se doori kitni hai? Jawab: 25 metre (tan 45° = 1).
4. Ek tower ki chhaya uski unchai ki √3 guna hai. Surya ka unnayan kon kitna hai? Jawab: 30° (tan θ = unchai / chhaya = 1/√3).
5. Ek hi taraf ke do binduon se ek tower ke shikhar ke unnayan kon 45° aur 60° hain, aur dono binduon ke beech ki doori 30 metre hai. Tower ki unchai kitni hai? Jawab: lagbhag 70.98 metre. Hal: x√3 = x + 30 se x = 15(√3 + 1) = 40.98, aur h = x√3 = 15(3 + √3) = 70.98 metre.
In practice questions ko pehle khud hal karein, phir answers se milayein. Jo question galat ho, uska diagram dobara banayein aur dekhein ki ratio chunne me kahan galti hui. Trikonmiti ke prayog class 10 notes ka asli fayda tabhi hai jab aap har example ko haath se hal karein, sirf padhne se nahi.
Aksar Puche Jaane Wale Sawaal
Trikonmiti ke prayog class 10 notes me sabse zyada kaunsa ratio use hota hai?
Tan sabse zyada use hota hai, kyunki zyadatar sawalon me unchai aur doori di ya poochhi jaati hai. Jab karn (jaise patang ki dori ya seedhi) shamil ho to sin ya cos use hota hai.
Unnayan kon aur avnaman kon me kya farq hai?
Unnayan kon upar dekhne par banta hai, avnaman kon neeche dekhne par. Avnaman kon wale sawal me alternate interior angles ki wajah se neeche wale tribhuj me wahi kon istemal ho jaata hai.
Kya Chapter 9 ke liye Chapter 8 yaad hona zaroori hai?
Haan, bilkul. Chapter 9 me 30°, 45° aur 60° ke sin, cos, tan ke maan har sawal me chahiye hote hain. Agar ye maan yaad nahi hain to Chapter 9 ke sawal hal nahi honge.
Board exam me Chapter 9 se kitne marks ke sawal aate hain?
Aam taur par Chapter 9 se 3 se 5 marks ke sawal aate hain, kabhi 2 marks ka chhota sawal aur kabhi 5 marks ka lamba sawal. Isliye is chapter ko poori tarah taiyaar rakhna chahiye.
Do binduon wale sawal ka standard tareeka kya hai?
Do equations banayein — pehle bindu se ek equation, doosre bindu se doosri equation — aur dono me unchai ko barabar rakhein. Isse pehle bindu ki doori nikal aati hai, phir unchai nikal aati hai.
Aakhri Baat
Trikonmiti ke prayog class 10 notes ka poora raaz sirf teen cheezon me hai: saaf diagram, sahi ratio ki pehchan, aur standard angles ke maan. Jo student har sawal me pehle diagram banata hai, wo aadha sawal wahin hal kar leta hai. Upar diye gaye 6 solved examples aur 5 practice questions ko haath se hal karein, to board exam me heights-and-distances ka koi bhi sawal aapko mushkil nahi lagega. Agle notes me hum Chapter 11 (Areas Related to Circles) cover karenge.
Official Links
- NCERT Class 10 Mathematics, Chapter 9 — ncert.nic.in par e-pathshala section me
- CBSE Academic, Class 10 Maths Syllabus — cbseacademic.nic.in
Is post se judi doosri posts: Nirdeshank Jyamiti Class 10 Notes aur Prishthiya Kshetrafal Aur Ayatan Class 10 Notes.