Summary & JEE Exam Tips: Properties of Triangles — Class 11 Trigonometry Notes

Sine Rule: Relates sides to the sines of opposite angles and the circumradius (R).

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

Cosine Rule: Relates the three sides to the cosine of an angle. Essential for SSS and SAS cases.

$\cos A = \frac{b^2+c^2-a^2}{2bc}$

Projection Formula: Expresses a side in terms of the projections of the other two sides.

$a = b\cos C + c\cos B$

Area of a Triangle ( $\Delta$ ):
- $\Delta = \frac{1}{2}ab\sin C$
- $\Delta = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{(Heron's Formula)}$
Circles and Radii:
- Circumradius (R): $R = \frac{abc}{4\Delta}$
- Inradius (r): $r = \frac{\Delta}{s}$
- Exradii ( $r_1, r_2, r_3$ ): $r_1 = \frac{\Delta}{s-a}$ , etc.
Half-Angle Formulas: Express $\sin(A/2), \cos(A/2), \tan(A/2)$ in terms of the sides (s, a, b, c).

Know When to Use Which Rule: This is the most crucial skill.
- Sine Rule: Use for AAS, ASA, and the ambiguous SSA case.
- Cosine Rule: Use for SAS (finding the third side) and SSS (finding any angle).
Area is the Bridge: The area of a triangle ( $\Delta$ ) is the central quantity that connects sides, angles, the inradius, and the circumradius. Many problems are solved by calculating the area first using one formula (like Heron's) and then using it in another formula (like $R=abc/4\Delta$ or $r=\Delta/s$ ) to find the required quantity.
Master Half-Angle and Tangent Rules: For problems relating sides to complex angle combinations, half-angle formulas and Napier's Analogy ( $\tan(\frac{B-C}{2}) = ...$ ) are often the intended shortcuts.
Equilateral Triangle Properties: For an equilateral triangle with side 'a', memorize these results: $R=a/\sqrt{3}$ and $r=a/(2\sqrt{3})$ . This gives the very important ratio R = 2r.
Link to Other Chapters: Be prepared for problems that connect Properties of Triangles with other topics. For example, questions where sides or angles are in AP/GP are very common.