Trigonometry – All Basic Ratios & Identities | JEE Main Complete Notes
Trigonometry JEE Mathematics ka ek fundamental topic hai. Is chapter ke basic ratios aur identities aage ke topics jaise equations, functions, inverse trigonometry, heights and distances aur calculus ke questions mein repeatedly use hote hain.
Agar aapko sin, cos, tan, cosec, sec, cot ke relations, standard values, quadrants aur fundamental identities perfectly yaad hain, to bahut saare JEE Main questions ko directly simplify kiya ja sakta hai.
1. Right Triangle Basics ⭐
Trigonometric ratios ko samajhne ke liye sabse pehle right-angled triangle ko understand karna zaroori hai.
/|
/ |
/ | P
H / |
/ θ__|
B
Angle θ ke respect mein:
- P = Perpendicular / Opposite
- B = Base / Adjacent
- H = Hypotenuse
Right triangle mein hypotenuse hamesha 90° angle ke opposite hoti hai aur triangle ki longest side hoti hai.
SOH–CAH–TOA ⭐
Basic ratios yaad rakhne ka easiest shortcut hai:
SOH → Sin = Opposite / Hypotenuse
CAH → Cos = Adjacent / Hypotenuse
TOA → Tan = Opposite / Adjacent
Therefore:
sin θ = P/H
cos θ = B/H
tan θ = P/B
2. Six Basic Trigonometric Ratios
Trigonometry mein total six basic ratios hote hain.
Primary Ratios
| Ratio | Formula |
|---|---|
| sin θ | P/H |
| cos θ | B/H |
| tan θ | P/B |
Reciprocal Ratios
| Ratio | Formula |
|---|---|
| cosec θ | H/P = 1/sin θ |
| sec θ | H/B = 1/cos θ |
| cot θ | B/P = 1/tan θ |
Important: sin, cos aur tan ko primary ratios samjho, while cosec, sec aur cot unke reciprocal ratios hain.
3. Reciprocal Relations 🔥
Reciprocal pairs ko ek line mein yaad rakho:
sin ↔ cosec
cos ↔ sec
tan ↔ cot
Isliye:
sin θ × cosec θ = 1
cos θ × sec θ = 1
tan θ × cot θ = 1
Example:
Agar sin θ = 3/5, then:
cosec θ = 5/3
Similarly, agar cos θ = 4/5, then:
sec θ = 5/4
4. Fundamental Trigonometric Identities ⭐
JEE Main mein trigonometric simplification ke liye sabse important identities Pythagorean identities hain.
Identity 1
sin²θ + cos²θ = 1
Isse hum directly obtain kar sakte hain:
sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
Identity 2
Above identity ko cos²θ se divide karne par:
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
Therefore:
tan²θ + 1 = sec²θ
Identity 3
Above identity ko sin²θ se divide karne par:
1 + cot²θ = cosec²θ
Identity Family 🔥
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
In teen identities ko JEE ke liye extremely important samjho.
5. Ratio Relations
Tan aur cot ko sin aur cos ke form mein likhna bahut useful hai.
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Example:
Agar sin θ = 3/5 aur cos θ = 4/5, then:
tan θ = (3/5)/(4/5) = 3/4
Similarly:
cot θ = 4/3
6. Important Derived Identities
Reciprocal relations se directly kuch useful formulas milte hain:
sin θ = 1/cosec θ
cos θ = 1/sec θ
tan θ = 1/cot θ
Ratio form:
sin θ / cos θ = tan θ
cos θ / sin θ = cot θ
JEE questions mein complicated expressions ko simple banane ke liye ye conversions extremely useful hain.
7. Complementary Angle Identities 🔥
Agar do angles ka sum 90° hai, to unhe complementary angles kehte hain.
A + B = 90°
Most important rule:
90° − θ → Ratio changes to its co-function
| Identity | Result |
|---|---|
| sin(90° − θ) | cos θ |
| cos(90° − θ) | sin θ |
| tan(90° − θ) | cot θ |
| cot(90° − θ) | tan θ |
| sec(90° − θ) | cosec θ |
| cosec(90° − θ) | sec θ |
Example:
sin 60° = cos 30°
Similarly:
tan 60° = cot 30°
8. Standard Trigonometric Values ⭐
JEE Main mein 0°, 30°, 45°, 60°, 90° ke standard values bahut frequently use hote hain.
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Not Defined |
| cosec θ | ND | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | ND |
| cot θ | ND | √3 | 1 | 1/√3 | 0 |
ND = Not Defined
Easy Memory Trick
Sin ke values:
0 → 1/2 → 1/√2 → √3/2 → 1
Cos ke values simply reverse order mein yaad kiye ja sakte hain:
1 → √3/2 → 1/√2 → 1/2 → 0
Tan ko directly:
tan θ = sin θ / cos θ
se calculate kar sakte ho.
9. Signs in Four Quadrants – ASTC Rule 🔥
Trigonometric ratios ka sign angle ke quadrant par depend karta hai.
Shortcut hai:
ASTC
A → All
S → Sin
T → Tan
C → Cos
| Quadrant | Positive Ratios |
|---|---|
| I | All |
| II | sin, cosec |
| III | tan, cot |
| IV | cos, sec |
Memory: All → Sin → Tan → Cos
Reciprocal ratio ka sign bhi uske corresponding primary ratio ke same hota hai.
10. Negative Angle Identities
Negative angles ke case mein kuch functions even hote hain aur kuch odd.
Even Functions
cos(−θ) = cos θ
sec(−θ) = sec θ
Odd Functions
sin(−θ) = −sin θ
cosec(−θ) = −cosec θ
tan(−θ) = −tan θ
cot(−θ) = −cot θ
Memory: sin, tan, cot → odd; cos, sec → even.
11. Domain & Range ⭐
Basic trigonometric functions ke domain aur range ko understand karna functions ke questions mein important hai.
| Function | Range | Important Restriction |
|---|---|---|
| sin θ | −1 ≤ sin θ ≤ 1 | None |
| cos θ | −1 ≤ cos θ ≤ 1 | None |
| tan θ | All real numbers | θ ≠ (2n+1)90° |
| sec θ | sec θ ≤ −1 or sec θ ≥ 1 | cos θ ≠ 0 |
| cosec θ | cosec θ ≤ −1 or cosec θ ≥ 1 | sin θ ≠ 0 |
| cot θ | All real numbers | θ ≠ nπ |
Yahan n generally integer hota hai.
12. When Is a Trigonometric Ratio Not Defined? ⚠️
Undefined values ko understand karne ke liye denominator check karo.
Tan θ
tan θ = sin θ / cos θ
Therefore, agar cos θ = 0, tan undefined hoga.
Examples: 90°, 270°, ...
Cot θ
cot θ = cos θ / sin θ
Agar sin θ = 0, cot undefined hoga.
Examples: 0°, 180°, 360°, ...
Sec θ
sec θ = 1/cos θ
Therefore:
cos θ = 0 → sec θ is undefined
Cosec θ
cosec θ = 1/sin θ
Therefore:
sin θ = 0 → cosec θ is undefined
13. Triangle Shortcut 🔥
JEE questions mein agar tan θ diya ho, to ek reference triangle immediately construct karna bahut useful hota hai.
Suppose:
tan θ = 3/4
Since:
tan θ = P/B
Take:
P = 3k
B = 4k
Pythagoras theorem:
H² = P² + B²
Therefore:
H = 5k
Now all ratios can be obtained:
sin θ = 3/5
cos θ = 4/5
tan θ = 3/4
cosec θ = 5/3
sec θ = 5/4
cot θ = 4/3
Ye famous 3–4–5 triangle JEE Main mein bahut useful shortcut hai.
14. Identity Simplification Strategy
JEE mein identity-based expression dekhkar random formulas apply mat karo. Ek systematic approach follow karo.
Step 1 → Everything into sin and cos
For example:
tan θ → sin θ/cos θ
cot θ → cos θ/sin θ
sec θ → 1/cos θ
cosec θ → 1/sin θ
Step 2 → Fundamental Identity Use Karo
sin²θ + cos²θ = 1
Step 3 → Required Pythagorean Identity Apply Karo
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Step 4 → LCM / Factorization
Expression ko common denominator mein convert karke simplify karo.
15. Important JEE Patterns 🔥
Pattern 1
1 − sin²θ = cos²θ
Pattern 2
1 − cos²θ = sin²θ
Pattern 3
sec²θ − tan²θ = 1
Pattern 4
cosec²θ − cot²θ = 1
Pattern 5
Consider:
tan θ + cot θ
Convert into sin and cos:
sin θ/cos θ + cos θ/sin θ
Taking LCM:
(sin²θ + cos²θ)/(sinθ cosθ)
Using:
sin²θ + cos²θ = 1
We get:
tan θ + cot θ = 1/(sinθ cosθ)
16. JEE Main-Level Solved Question 🔥
Question
If:
tan θ = 3/4
and θ lies in the third quadrant, find:
sin θ + cos θ
Step 1 → Reference Triangle
Given:
tan θ = P/B = 3/4
Therefore:
P = 3, B = 4, H = 5
Hence magnitudes are:
|sin θ| = 3/5
|cos θ| = 4/5
Step 2 → Third Quadrant Sign
ASTC rule ke according QIII mein:
tan positive
But:
sin θ < 0
cos θ < 0
Therefore:
sin θ = −3/5
cos θ = −4/5
Step 3 → Required Expression
sin θ + cos θ = −3/5 − 4/5
Answer = −7/5 ✅
JEE Lesson: Ratio se magnitude milta hai, lekin final sign quadrant decide karta hai.
17. Common JEE Traps ⚠️
❌ Wrong: tan θ = cos θ/sin θ
✅ Correct: tan θ = sin θ/cos θ
❌ Wrong: sec θ = 1/sin θ
✅ Correct: sec θ = 1/cos θ
❌ Wrong: All ratios are positive in every quadrant.
✅ Correct: Use the ASTC rule.
❌ Wrong: tan 90° = 0
✅ Correct: tan 90° is Not Defined.
❌ Wrong: cosec 0° = 0
✅ Correct: cosec 0° is Not Defined.
❌ Wrong: sin²θ means sin(2θ)
✅ Correct: sin²θ = (sin θ)²
18. JEE Main Quick Solving Strategy
Trigonometry ka basic question dekhte hi ye sequence follow karo:
FIRST → RATIO → QUADRANT → IDENTITY → SIMPLIFY
FIRST: Given information identify karo.
RATIO: sin, cos, tan etc. ke relations use karo.
QUADRANT: Sign check karo using ASTC.
IDENTITY: Required fundamental identity apply karo.
SIMPLIFY: LCM, factorization ya cancellation se answer nikalo.
Is approach se unnecessary calculations avoid ho sakti hain.
19. One-Minute Revision ⚡
SOH–CAH–TOA:
sin = P/H
cos = B/H
tan = P/B
Reciprocals:
cosec = H/P
sec = H/B
cot = B/P
Ratio Relations:
tan = sin/cos
cot = cos/sin
Main Identities:
sin² + cos² = 1
1 + tan² = sec²
1 + cot² = cosec²
Complement:
sin ↔ cos
tan ↔ cot
sec ↔ cosec
Quadrants:
All → Sin → Tan → Cos
Triangle:
tan θ = P/B
3–4–5 triangle:
P = 3, B = 4, H = 5
20. Final Revision Box 🔥
Six Ratios
sin = P/H
cos = B/H
tan = P/B
cosec = H/P
sec = H/B
cot = B/P
Three Main Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Ratio
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ
Complement
sin ↔ cos
tan ↔ cot
sec ↔ cosec
Quadrants
A → S → T → C
All → Sin → Tan → Cos
Triangle
tanθ = P/B
P² + B² = H²
Golden Rule
FIRST → RATIO → QUADRANT → IDENTITY → SIMPLIFY
```21. Practice Questions for JEE Main
Q1. If tan θ = 5/12 and θ lies in the first quadrant, find sin θ and cos θ.
Q2. Simplify: 1 − sin²θ.
Q3. Simplify: sec²θ − tan²θ.
Q4. If sin θ = 3/5 and θ lies in the second quadrant, find cos θ.
Q5. Find the value of sin(90° − 30°).
Q6. Simplify: tan θ + cot θ.
Q7. Which trigonometric ratios are positive in the third quadrant?
Q8. Find the value of cosec 30° + sec 60°.
22. PDF – Download / Study Notes
Below you can embed the complete PDF notes for students who want to revise the topic offline.
23. Frequently Asked Questions – FAQs
Q1. Trigonometry mein sabse pehle kya yaad karna chahiye?
Sabse pehle SOH–CAH–TOA, six ratios aur teen fundamental identities yaad karo. Uske baad standard values aur ASTC rule master karo.
Q2. JEE Main mein trigonometric identities important hain?
Yes. Basic identities bahut important hain kyunki ye direct identity questions ke saath-saath algebraic simplification aur higher-level trigonometric problems mein bhi repeatedly use hoti hain.
Q3. tan θ ko sin aur cos mein kaise likhen?
tan θ = sin θ/cos θ
Q4. Third quadrant mein kaunse ratios positive hote hain?
ASTC ke according third quadrant mein tan aur cot positive hote hain.
Q5. Kya tan 90° defined hai?
No. Since tan θ = sin θ/cos θ and cos 90° = 0, therefore tan 90° is not defined.
Q6. 3–4–5 triangle ka use kyun important hai?
Agar question mein tan θ = 3/4 type ratio diya ho, to immediately P = 3, B = 4 aur H = 5 assume karke baaki ratios quickly obtain kiye ja sakte hain.
Q7. sin²θ ka meaning kya hai?
sin²θ = (sin θ)². Ye sin(2θ) nahi hota.
Q8. Complementary angle identities ka shortcut kya hai?
90° − θ dekhte hi ratio ko uske co-function mein change karo: sin ↔ cos, tan ↔ cot, sec ↔ cosec.
24. Final Thoughts
Trigonometry ke basic ratios aur identities JEE Mathematics ki foundation hain. Is topic ko sirf formulas ratne ke bajay right triangle + ratio + identity + quadrant ke logic se samajhna chahiye.
Agar aapko six ratios, SOH–CAH–TOA, three Pythagorean identities, standard values, complementary identities, ASTC rule aur 3–4–5 triangle shortcut confidently aata hai, to basic trigonometry ke majority questions ka approach clear ho jata hai.
JEE Main ke questions solve karte waqt ek golden sequence yaad rakho:
FIRST → RATIO → QUADRANT → IDENTITY → SIMPLIFY
Is sequence ko practice mein repeatedly apply karo. Trigonometry ka confidence formulas ki quantity se nahi, balki correct formula ko correct situation mein identify karne se aata hai.