Class 11 Maths NCERT Solutions: Trigonometric Functions

Chapter 3 - Samikaran aur Uttar

Trigonometric Functions ka ye chapter Class 11 ke liye koi mamuli nahi hai, yaar—bahut heavy hai. Isme hum angles aur triangles ke beech ka connection samjhenge, woh bhi aise ki base clear ho jaye. Students ko yahaan kya milega?

• Basic trigonometric ratios aur unki identities—yeh chapter ka sabse zaroori hissa hai. Samikaran aur uttar ke beech ka pul yahi hai. Jab tak aap ratios ko clearly nahi samajhte, identities se uttar nikalna mushkil ho jata hai. Isliye pehle ratios ko theek se pakdo, phir identities apne aap asaan lagne lagengi. Kuch log inhe ratne ki koshish karte hain, lekin samajhna behtar hai—ratna aaj kaam aata hai, kal nahi. To seedha seedha dekhein: ratios base hain, identities unke upar bani hain. Aur uttar? Woh toh inhi dono ke mel se milta hai.
• That heading isn't a paragraph—it's just two words. You've given me a section title, not body text to work with. Drop the actual paragraph content in here and I'll rewrite it for you, keeping it tight and natural under that Chapter 3 heading.
• Trigonometric functions ke graphs—where do I even start with these? Honestly, they’re not as scary as they look. Once you get the hang of them, they’re just waves doing their thing, repeating over and over. Sine and cosine? They’re like the classic duo, smooth and predictable, rolling up and down forever. Tangent, though, is a whole different beast—it shoots off to infinity, comes back, and repeats, all while looking a little chaotic. But that’s the charm of it. You plot a few points, connect them with a curve, and suddenly you’ve got a shape that just makes sense. The x-axis is your stage, the y-axis is your range, and the graph just dances between the peaks and valleys. Nothing crazy, just patterns. And once you see the pattern, you can’t unsee it.
• You can’t just memorize these and hope for the best. You have to actually know where each one fits, and more importantly, when to pull it out. Because here’s the thing—formulas are only half the battle. The real trick is figuring out which equation matches the problem staring back at you, and that takes a bit of practice, honestly. Some will feel obvious right away, others will make you second-guess everything. But once you start spotting the patterns, it clicks. You just have to get your hands dirty with them first.

Har concept ko samajhna ab bilkul aasaan ho jaayega, kyunki niche diye gaye solutions bilkul detail mein hain. Bas inhe dhyan se padhiye, sab kuch clear ho jayega.

Links for Chapter-wise Download NCERT Solution for Class 11 गणितII in hindi Language

Here we have provided NCERT Solution for Class 11 गणितII in hindi Language, Just select the chapters below to get solution of the same:

शांकव परिच्छेदन

त्रिविमीय ज्यामिति का परिचय

सीमा और अवकलज

गणितीय विवेचन

सांख्यिकी

प्रायिकता

Trigonometric Functions - Complete NCERT Solutions

Yeh solutions specifically Class 11 Maths ke Chapter 3 ke liye banaye gaye hain. Har ek question ko step-by-step todha gaya hai, taaki students concept ko seedha apne dimaag mein utaar saken—koi confusion ka scope nahi. Aur haan, solutions bilkul Hinglish mein hain, isliye padhne mein koi atak nahi aati, sab kuch naturally samajh aa jata hai.

1. Basic Concepts aur Definitions

Sabse pehle baat karte hain trigonometric ratios ki. Ek right-angled triangle mein, bas teen basic ratios hote hain — sine, cosine, aur tangent. Aur inhe define karte hain is tarah se:

• θ isn’t some abstract mystery — it’s a ratio, plain and simple. Take the side directly across from your angle—that’s the opposite side. Then grab the longest side of the triangle, the hypotenuse. Divide the first by the second. That’s it — no tricks, no hidden steps. Just sin θ equals opposite over hypotenuse, and honestly, once you see it that way, it sticks.
• Alright, so first things first, you gotta know what this little guy means. θ is just the angle we're looking at. Now, for that angle, we care about two sides of the triangle. The adjacent side is the one that's right next to θ, and it's not the hypotenuse—that's the longest side, always opposite the right angle. So, you take that adjacent side, you divide it by the hypotenuse. Boom, you get cos θ. It's a ratio, pure and simple. No magic, just a way to describe that angle using the triangle's proportions.
• θ is all about the ratio between two sides of a right triangle. You take the side straight across from the angle—that’s your opposite—and you divide it by the side that’s right next to θ. Not the hypotenuse. That’s the adjacent side. So you just do opposite over adjacent, and boom, you’ve got tan θ. Simple as that, once you spot which side is which.

Inhi basic ratios se hi hum aage chalke cosec, sec, aur cot nikaal lete hain. Aur dekho, ye saare ratios aapas mein ek doosre se is tarah juday hote hain ki inke beech trigonometric identities ka ek poora jaal hai, jo inhe ek sutra mein bandh deta hai.

2. Degree aur Radian Measure

Angles ko naapne ke do basic tarike hain—degrees aur radians. Socho, ek poora circle 360 degrees ka hota hai, ya agar radians ki baat karein, toh wohi circle 2π radians ke barabar hai. Ab sawaal ye hai ki in dono ke beech kaise convert karein? Formula simple hai: radians = (π/180) × degrees. Bas isi formula ko haath mein lo, aur aap problems solve karne lag jao. Yehi woh chabi hai jo kaam aati hai.

Example: 60 degrees ko radians mein convert karte hain. Solution: Radians = (π/180) × 60 = π/3 radians. Wait, let me redo that—simple formula, straight plug-in. Take 60, multiply by π/180, and boom, you get π/3. That’s it. No tricks, no extra steps. The conversion’s clean: degrees shrink down to a neat fraction of pi. So 60° equals π/3 radians, and you’re done.

3. Trigonometric Functions ke Graphs

Har trigonometric function ka apna alag style hai—bilkul apni pehchan wala graph. Sine aur cosine dono periodic hain, aur inka period 2π hai. Matlab, har 2π ke baad pattern repeat hota hai. Lekin tangent ka scene thoda hatke hai; uska period sirf π hai. In graphs ko padhna seekhna zaroori hai, kyunki ye sach mein bata dete hain ke function kis tarah behave karta hai. Ek baar inhe samajh lo, toh aage ke sawal easy lagne lagte hain.

Plotting graphs mein sabse pehle domain aur range ka khayal rakhna padta hai—yehi base hai. Dekho, sine function ki range sirf -1 se 1 ke beech bandh hoti hai, isse zyada upar ya neeche ja nahi sakti.

4. Important Trigonometric Identities

Okay, let’s be real. There are a few basic identities you just can’t afford to forget. Seriously, they pop up everywhere, so get them locked in your memory.

• There’s a reason this one gets called the Pythagorean identity—it’s basically the Pythagorean theorem hiding inside a circle. Think about it: for any angle θ, if you square the sine and the cosine, they always add up to exactly 1. No exceptions. That’s not a coincidence, that’s geometry doing its thing. It’s also the identity you’ll reach for more than any other. Need to swap a sin² for 1 − cos²? Done. Or maybe turn a cos² into 1 − sin²? Same trick, works every time. Once you get comfortable with this one, a lot of other trig problems start to feel a whole lot less scary.
• Okay, so here's the thing about this one. You can't just memorize it and call it a day, you have to see it as a direct consequence of the most fundamental identity out there: sin²θ + cos²θ = 1. Just take that bad boy and divide every single term by cos²θ. That's the whole trick. Watch what happens. You get sin²θ/cos²θ, which is just tan²θ, then you get cos²θ/cos²θ which simplifies down to a nice, clean 1. On the other side you've got 1/cos²θ, which is, by definition, sec²θ. So it all just falls right into place. It's not some random, unrelated formula you have to blindly trust; it's literally a rearranged version of the Pythagorean identity. Once that clicks, you'll never have to force yourself to remember it again.
• Here’s the rewritten version: This one’s a bit of a twist on the others, but it’s just as handy. Start with 1 + cot²θ and you land on cosec²θ. Honestly, it’s less intuitive at first glance, so it’s worth drilling until it sticks. The logic flows straight from the earlier identities, but you’ll save yourself a headache if you just memorize it cold.

Honestly, practice is where these formulas really stick. You don’t just read them once and magically remember—no way. You grind through problems, mess up a few times, and then one day they’re just there, automatic. That’s the trick with trigonometric identities: they’re not meant to be memorized in a vacuum. You use them to break down gnarly expressions into something manageable. So yeah, keep solving, keep simplifying, and soon you won’t even think twice—you’ll just see the pattern and go.

5. Practice Problems with Solutions

Ab hum kuch important NCERT problems utha rahe hain, aur unke solutions bhi dekh lenge. Seedha seedha, step by step — koi jhanjhat nahi, bas practice.

Problem: Agar sin A = 3/5 hai, toh cos A aur tan A ki value nikaalni hai. Solution: Sabse pehle, humein pata hai ki sin²A + cos²A = 1 hota hai. Toh ab isse hum cos²A ki value nikaal sakte hain: cos²A = 1 - sin²A = 1 - (9/25) = 16/25 Ab agar hum square root lein, toh cos A = 4/5 milega. Aur haan, hum positive sign lenge kyunki question mein diya hai ki A first quadrant mein hai, wahan cos hamesha positive hota hai. Aakhri mein, tan A ki baat karein toh tan A = sin A / cos A = (3/5) / (4/5) = 3/4. Bas, yahi answer hai.

6. Tips for Exam Preparation

• Sabse pehle, all basic formulas yaad karein. Honestly, this is where you gotta start—no way around it. Get those fundamentals locked in your head first. Sure, it feels tedious, but everything else builds on them. So don’t skip this step. Drill them until they’re second nature, like muscle memory for your brain. That’s the solid ground you need before tackling anything trickier.
• Practice regularly, but don’t just stick to the same kind of problems—mix it up. Solve different types, throw in some tricky ones, maybe a few you’ve never seen before. That’s where the real learning kicks in. It keeps your brain on its toes, and honestly, it makes the whole thing less boring. You’re not just repeating steps; you’re actually getting comfortable with whatever the exam might throw at you. So, grab a mix, work through them, and watch your confidence climb.
• Look, if you really want to crack an exam, there’s one habit that beats almost everything else—solving previous years’ question papers. And I mean actually sitting down with them, not just glancing through. That’s where the real gold is hiding. It shows you the pattern, the kind of questions that repeat, the tricky spots teachers love to hit. You start to see the exam’s personality, if that makes sense. Plus, it’s the fastest way to kill that “what if I freeze on the big day” anxiety. So grab those old papers, time yourself like it’s the real deal, and go. You’ll thank yourself later.
• Graphs can be tricky, so don’t just read them—actually get your hands dirty and practice plotting them yourself. Scribble them out on rough paper, label the axes, check the scales. It’s one thing to see a graph in a book and think you’ve got it. Another to draw it from scratch when you’re under time pressure. So, grab a pencil and plot a few until the process feels almost automatic. That little bit of extra practice could save you in the exam hall.
• Got a doubt — don’t let it sit there. Go ask your teacher right away, or flip through a reference book and sort it out on the spot.

Solution mein trigonometric functions ke concepts apne aap clear ho jaate hain, bas ek baar dhang se dekho toh. Har step ko dhyan se follow karo, aur phir khud se practice karo — wahi asli baat hai. Thoda sa confidence ke saath exam ki taiyaari karo, aur dekho kya farak padta hai.