TrigonometryHigh Yield (1 Q in JEE)

Properties of Triangles (SOT) Formulas

Sine rule, cosine rule, projection formula, radii of circumcircle, incircle, and excircles.

8 Formulas•351+ Questions in Bank

Core Formulas & Mathematical Identities

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, \quad R = \frac{abc}{4\Delta}

Connects sides, angles, and circumradius R of triangle ABC.

Examiner Tip: Δ is the area of triangle: Δ = (1/2)ab sin C.

\cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}

Determines angles when all three sides are known.

Examiner Tip: If a² = b² + c², then A = 90° (Pythagoras theorem).

r = \frac{\Delta}{s} = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \quad r_1 = \frac{\Delta}{s - a} = 4R\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}

Radii of incircle and excircles (opposite to vertices A, B, C).

Examiner Tip: 1/r₁ + 1/r₂ + 1/r₃ = 1/r and r₁ r₂ r₃ = s Δ.

\tan\left(\frac{B - C}{2}\right) = \frac{b - c}{b + c}\cot\frac{A}{2}, \quad \tan\left(\frac{C - A}{2}\right) = \frac{c - a}{c + a}\cot\frac{B}{2}, \quad \tan\left(\frac{A - B}{2}\right) = \frac{a - b}{a + b}\cot\frac{C}{2}

Connects difference of two angles to the ratio of side lengths and half the third angle.

Examiner Tip: Often utilized in JEE Advanced when side differences (b - c) or angle differences are given.

\sin\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}, \quad \cos\frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}}, \quad \tan\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} = \frac{\Delta}{s(s-a)}

Trigonometric half-angle ratios expressed purely in terms of side lengths and semi-perimeter s.

Examiner Tip: Area of triangle via Heron's formula is Δ = √(s(s-a)(s-b)(s-c)).

OI^2 = R^2 - 2Rr \; (\text{Euler's Incenter Formula}), \quad HG : GO = 2 : 1 \; (\text{Euler Line})

Metric distance between circumcenter O and incenter I, and collinearity ratio of Orthocenter (H), Centroid (G), and Circumcenter (O).

Examiner Tip: In an equilateral triangle, H, G, I, and O all coincide into a single point, giving R = 2r.

a = b\cos C + c\cos B, \quad m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}, \quad m_a^2 + m_b^2 + m_c^2 = \frac{3}{4}(a^2 + b^2 + c^2)

Projection of sides onto each other, and closed-form length of median from vertex A bisecting opposite side a.

Examiner Tip: Centroid divides each median in 2:1 ratio from vertex: AG = (2/3) m_a.

(m + n)\cot\theta = m\cot\alpha - n\cot\beta = n\cot B - m\cot C

Relates angle θ formed by a cevian dividing the base into segments m : n to the base angles or divided vertex angles α and β.

Examiner Tip: A staple shortcut in JEE Advanced geometry questions involving internal transversals.

Frequently tested misconceptions in Properties of Triangles (SOT)