Trigonometric Functions
Trigonometry me hum angles aur unse jude ratios padhte hain. Class 10 wali "right-triangle trigonometry" ko yahan hum general angles tak badhate hain — unit circle ki madad se. Ye chapter angles ki measurement, trig functions, unki identities aur equations cover karta hai.
1. Angle Measurement: Degree aur Radian
Angle do tarah se naapte hain:
- Degree: ek full circle \(= 360^\circ\); \(1^\circ = 60'\) (minutes), \(1' = 60''\) (seconds).
- Radian: ek circle ke center par wo angle jo radius ke barabar arc banata hai \(= 1\) radian.
Degree aur radian ka relation: \[ \pi \text{ radian} = 180^\circ. \] Isse: \(1 \text{ radian} = \dfrac{180^\circ}{\pi} \approx 57^\circ 16'\), aur \(1^\circ = \dfrac{\pi}{180}\) radian.
Conversion ka rule:
\[ \text{Radian} = \frac{\pi}{180} \times \text{Degree}, \qquad \text{Degree} = \frac{180}{\pi} \times \text{Radian}. \]
Arc length
Radius \(r\) wale circle me, \(\theta\) radian ke angle ka arc length:
\[ l = r\,\theta. \]
\(40^\circ\) ko radian me badlo. \[ 40^\circ = \frac{\pi}{180} \times 40 = \frac{2\pi}{9} \text{ radian}. \]
2. Trigonometric Functions (Unit Circle)
Origin par center wale unit circle (\(r = 1\)) par, \(x\)-axis se \(\theta\) angle ghoomne par jo point \(P(x, y)\) milta hai, uske coordinates se:
\[ \cos\theta = x, \qquad \sin\theta = y. \]
Baaki functions inhi se bante hain:
\[ \tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}. \]
Signs in Quadrants (ASTC)
Yaad rakhne ka trick — "All Silver Tea Cups":
- Quadrant I: All positive.
- Quadrant II: Sin (aur cosec) positive.
- Quadrant III: Tan (aur cot) positive.
- Quadrant IV: Cos (aur sec) positive.
Domain aur Range
| Function | Domain | Range |
|---|---|---|
| \(\sin x,\ \cos x\) | \(\mathbb{R}\) | \([-1, 1]\) |
| \(\tan x\) | \(x \neq (2n+1)\frac{\pi}{2}\) | \(\mathbb{R}\) |
| \(\cot x\) | \(x \neq n\pi\) | \(\mathbb{R}\) |
| \(\sec x\) | \(x \neq (2n+1)\frac{\pi}{2}\) | \((-\infty,-1]\cup[1,\infty)\) |
| \(\csc x\) | \(x \neq n\pi\) | \((-\infty,-1]\cup[1,\infty)\) |
Sab functions periodic hain: \(\sin, \cos, \sec, \csc\) ka period \(2\pi\); \(\tan, \cot\) ka period \(\pi\).
3. Fundamental Identities
\[ \sin^{2}\theta + \cos^{2}\theta = 1 \]
\[ 1 + \tan^{2}\theta = \sec^{2}\theta, \qquad 1 + \cot^{2}\theta = \csc^{2}\theta \]
Even–odd: \(\cos(-x) = \cos x\) (even), \(\sin(-x) = -\sin x\), \(\tan(-x) = -\tan x\) (odd).
4. Sum aur Difference Formulas
\[ \cos(x + y) = \cos x \cos y - \sin x \sin y \]
\[ \cos(x - y) = \cos x \cos y + \sin x \sin y \]
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]
\[ \sin(x - y) = \sin x \cos y - \cos x \sin y \]
\[ \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}, \qquad \tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \]
\(\cos 75^\circ\) nikaalo. \[ \cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}}. \]
5. Multiple Angle Formulas
\[ \sin 2x = 2 \sin x \cos x, \qquad \cos 2x = \cos^{2} x - \sin^{2} x = 1 - 2\sin^{2} x = 2\cos^{2} x - 1 \]
\[ \tan 2x = \frac{2\tan x}{1 - \tan^{2} x}, \qquad \sin 3x = 3\sin x - 4\sin^{3} x, \qquad \cos 3x = 4\cos^{3} x - 3\cos x \]
6. Product ↔ Sum Formulas
\[ 2\cos x \cos y = \cos(x - y) + \cos(x + y) \]
\[ 2\sin x \sin y = \cos(x - y) - \cos(x + y) \]
\[ 2\sin x \cos y = \sin(x + y) + \sin(x - y) \]
7. Trigonometric Equations (General Solutions)
Yahan \(n \in \mathbb{Z}\) (koi bhi integer):
\[ \sin x = 0 \;\Rightarrow\; x = n\pi \]
\[ \cos x = 0 \;\Rightarrow\; x = (2n + 1)\frac{\pi}{2} \]
\[ \sin x = \sin y \;\Rightarrow\; x = n\pi + (-1)^{n} y \]
\[ \cos x = \cos y \;\Rightarrow\; x = 2n\pi \pm y \]
\[ \tan x = \tan y \;\Rightarrow\; x = n\pi + y \]
Key Takeaways
- \(\pi\) radian \(= 180^\circ\); arc length \(l = r\theta\).
- Unit circle: \(\cos\theta = x\), \(\sin\theta = y\); signs ke liye ASTC rule.
- Core identity: \(\sin^{2}\theta + \cos^{2}\theta = 1\).
- Sum/difference aur multiple-angle formulas problems ka base hain — yaad rakho.
- General solutions integer \(n\) ke saath aate hain.