Properties of Triangles & Heights-Distances

Properties of triangles connect trigonometric functions to triangle geometry through fundamental laws and formulas. The Sine Rule states a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. The Cosine Rule gives cos A = ($b^{2}$ + $c^{2}$ - $a^{2}$)/(2bc), providing a direct relationship between sides and angles. The Projection Formula gives a = bcos C + ccos B, useful in proving identities involving mixed side-angle expressions.

The area of a triangle has multiple trigonometric expressions: $\Delta$ = ($\frac{1}{2}$)absin C = ($\frac{1}{2}$)bcsin A = ($\frac{1}{2}$)casin B.
Heron's formula gives $\Delta$ = $\sqrt{s(s-a}$(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter.
The relationship $\Delta$ = rs (where r is the inradius) and $\Delta$ = abc/(4R) connect area to the special radii.

The inradius r = $\Delta$/s = (s-a)tan(A/2) = 4Rsin(A/2)*sin(B/2)*sin(C/2).
The exradii opposite to vertices A, B, C are r1 = $\Delta$/(s-a), r2 = $\Delta$/(s-b), r3 = $\Delta$/(s-c). These formulas enable JEE problems that relate the incircle and excircles to triangle dimensions.

Half-angle formulas are derived from the cosine rule: sin(A/2) = $\sqrt{(s-b}$(s-c)/(bc)), cos(A/2) = $\sqrt{s(s-a}$/(bc)), and tan(A/2) = $\sqrt{(s-b}$(s-c)/(s(s-a))) = $\Delta$/(s(s-a)). These are essential for problems involving half-angles and the inradius/exradius.

Napier's analogy (tangent rule) states tan((A-B)/2) = ((a-b)/(a+b)) * cot(C/2), useful when two sides and the included angle are known and you need the difference of the other two angles.

Heights and distances problems apply trigonometry to real-world scenarios. The angle of elevation is measured upward from the horizontal, and the angle of depression downward. Key techniques include: (1) drawing a clear diagram, (2) identifying right triangles, (3) using tan for height-distance problems (most common), (4) applying the sine rule for non-right triangle configurations, and (5) using the identity that the angle subtended by an object decreases as you move farther away.

The key problem-solving concept is selecting the appropriate formula (sine rule, cosine rule, area formula, or half-angle formula) based on the given information and the quantity to be determined.