Class 10 mein right triangle mein sin/cos/tan seekha tha — ab Class 11 mein yeh functions poore number line par extend ho jaate hain!
Socho ek wheel ek circle par ghoom raha hai. Agar us wheel ki height ko time ke saath plot karo, toh sine wave banta hai! Yahi reason hai ki trigonometric functions physics, music, aur engineering mein everywhere hain. Class 10 mein humne sirf 0°–90° ke angles ke liye trig seekha tha. Ab Class 11 mein hum saare angles ke liye — negative angles bhi, 360° se bade angles bhi — trig extend karte hain. Iske liye Unit Circle ka concept use karte hain.
Degree ek arbitrary unit hai (360 kyon? Babylonians ne choose kiya tha!). Radian ek natural unit hai: 1 radian = angle jab arc length = radius. Yahi reason hai ki advanced maths mein hamesha radians use hote hain — formulas simpler hote hain.
Conversion trick: Degree → Radian: ×(π/180) | Radian → Degree: ×(180/π)
| Degree | Radian | Degree | Radian |
|---|---|---|---|
| 0° | 0 | 180° | π |
| 30° | π/6 | 270° | 3π/2 |
| 45° | π/4 | 360° | 2π |
| 60° | π/3 | −90° | −π/2 |
| 90° | π/2 | −180° | −π |
Unit circle ka radius = 1 hota hai aur center origin par. Kisi bhi angle θ ke liye, circle par point P = (cos θ, sin θ) hota hai. Yahi definition hai Class 11 mein — right triangle nahi, unit circle!
Unit Circle: radius = 1. Koi bhi angle θ ke liye P = (cos θ, sin θ)
P = (x, y) unit circle par hai, angle θ ke liye — toh 6 trig functions yahan se define hote hain:
y/r = y
"Perpendicular/Hypotenuse" — y-coordinate
x/r = x
"Base/Hypotenuse" — x-coordinate
y/x = sin/cos
"P/B" — slope of radius
1/sin θ
sin ka reciprocal. Defined jab sin≠0
1/cos θ
cos ka reciprocal. Defined jab cos≠0
1/tan θ = cos/sin
tan ka reciprocal. Defined jab tan≠0
| θ | sin θ | cos θ | tan θ | cosec θ | sec θ | cot θ |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | ∞ | 1 | ∞ |
| 30° (π/6) | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° (π/4) | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° (π/2) | 1 | 0 | ∞ | 1 | ∞ | 0 |
| 180° (π) | 0 | −1 | 0 | ∞ | −1 | ∞ |
Har quadrant mein kaunse trig functions positive hain — yeh yaad karne ka trick hai "All Silver Tea Cups":
Q1: All (sin, cos, tan sab positive) | Q2: Sin positive (baaki negative) | Q3: Tan positive (baaki negative) | Q4: Cos positive (baaki negative)
Unit circle se directly — x²+y²=1. Sabse important identity!
Identity 1 ko cos²θ se divide karo.
Identity 1 ko sin²θ se divide karo.
Jab two angles ka sum ya difference ka trig value chahiye — yeh formulas kaam aate hain:
Trick: A+A = 2A — sum formula mein A=B rakh do!
Trig equations ke infinite solutions hote hain (kyunki sin/cos periodic hain with period 2π). General solution batata hai saare possible values:
Q1. 210° ko radians mein convert karo.
210 × π/180 = 7π/6 radians
Q2. Prove karo: (sin θ + cosec θ)² = sin²θ + cosec²θ + 2
LHS = sin²θ + 2sinθ·cosecθ + cosec²θ = sin²θ + 2sinθ·(1/sinθ) + cosec²θ = sin²θ + 2 + cosec²θ = RHS ✓
Q3. cos 15° ki value nikalo using cos(45°−30°).
cos 15° = cos45°cos30° + sin45°sin30° = (1/√2)(√3/2)+(1/√2)(1/2) = (√3+1)/(2√2) = (√6+√2)/4
Q4. tan²θ + 1 = sec²θ prove karo.
sin²θ + cos²θ = 1 ko cos²θ se divide karo → tan²θ + 1 = sec²θ ✓
Q5. sin θ = 1/2 ka general solution likho.
sin θ = sin(π/6) → θ = nπ + (−1)ⁿ(π/6), n ∈ ℤ