Introduction to Trigonometry: Key Concepts for CBSE Class 10
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles, particularly right-angled triangles. It has wide applications in various fields like engineering, physics, navigation, and astronomy.
1. Trigonometric Ratios
For an acute angle $\\theta$ in a right-angled triangle:
- Sine (sin $\\theta$): $\\frac{\\text{Opposite Side}}{\\text{Hypotenuse}}$
- Cosine (cos $\\theta$): $\\frac{\\text{Adjacent Side}}{\\text{Hypotenuse}}$
- Tangent (tan $\\theta$): $\\frac{\\text{Opposite Side}}{\\text{Adjacent Side}}$
- Cosecant (cosec $\\theta$): $\\frac{\\text{Hypotenuse}}{\\text{Opposite Side}} = \\frac{1}{\\text{sin } \\theta}$
- Secant (sec $\\theta$): $\\frac{\\text{Hypotenuse}}{\\text{Adjacent Side}} = \\frac{1}{\\text{cos } \\theta}$
- Cotangent (cot $\\theta$): $\\frac{\\text{Adjacent Side}}{\\text{Opposite Side}} = \\frac{1}{\\text{tan } \\theta}$
Mnemonic: SOH CAH TOA (Sine Opposite Hypotenuse, Cosine Adjacent Hypotenuse, Tangent Opposite Adjacent)
2. Trigonometric Ratios of Specific Angles
You must know the values for $0^\\circ, 30^\\circ, 45^\\circ, 60^\\circ,$ and $90^\\circ$.
| Angle ($\\theta$) | sin $\\theta$ | cos $\\theta$ | tan $\\theta$ | cosec $\\theta$ | sec $\\theta$ | cot $\\theta$ |
|---|---|---|---|---|---|---|
| $0^\\circ$ | 0 | 1 | 0 | Undefined | 1 | Undefined |
| $30^\\circ$ | $\\frac{1}{2}$ | $\\frac{\\sqrt{3}}{2}$ | $\\frac{1}{\\sqrt{3}}$ | 2 | $\\frac{2}{\\sqrt{3}}$ | $\\sqrt{3}$ |
| $45^\\circ$ | $\\frac{1}{\\sqrt{2}}$ | $\\frac{1}{\\sqrt{2}}$ | 1 | $\\sqrt{2}$ | $\\sqrt{2}$ | 1 |
| $60^\\circ$ | $\\frac{\\sqrt{3}}{2}$ | $\\frac{1}{2}$ | $\\sqrt{3}$ | $\\frac{2}{\\sqrt{3}}$ | 2 | $\\frac{1}{\\sqrt{3}}$ |
| $90^\\circ$ | 1 | 0 | Undefined | 1 | Undefined | 0 |
3. Trigonometric Identities
- sin$^2 \\theta +$ cos$^2 \\theta = 1$
- $1 +$ tan$^2 \\theta = $ sec$^2 \\theta$
- $1 +$ cot$^2 \\theta = $ cosec$^2 \\theta$
Also, remember:
- tan $\\theta = \\frac{\\text{sin } \\theta}{\\text{cos } \\theta}$
- cot $\\theta = \\frac{\\text{cos } \\theta}{\\text{sin } \\theta}$
4. Trigonometric Ratios of Complementary Angles
- sin $(90^\\circ - \\theta) = \\text{cos } \\theta$
- cos $(90^\\circ - \\theta) = \\text{sin } \\theta$
- tan $(90^\\circ - \\theta) = \\text{cot } \\theta$
- cot $(90^\\circ - \\theta) = \\text{tan } \\theta$
- sec $(90^\\circ - \\theta) = \\text{cosec } \\theta$
- cosec $(90^\\circ - \\theta) = \\text{sec } \\theta$
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