4. TRIGONOMETRY: SOLUTION OF TRIANGLES

Introduction — What Does “Solution of Triangles” Mean?

The solution of a triangle means finding all unknown sides and angles of a triangle when certain pieces of information are given. These formulas are useful when the triangle is not a right-angled triangle.

Typical problems involve finding missing sides, missing angles, or area of a triangle.

1. Sine Rule

For any triangle ABC:

a / sin A = b / sin B = c / sin C = 2R

Here, R is the circumradius (radius of the circle that passes through all three vertices).

Use sine rule when:

  • Two angles and one side (AAS or ASA) are known.
  • Two sides and a non-included angle (SSA) are known.

2. Cosine Rule

a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C

Use cosine rule when:

  • All three sides are known (SSS).
  • Two sides and included angle (SAS) are known.

3. Projection Rule

The projection rule relates sides with the cosine of angles:

b = a cos C + c cos B
c = a cos B + b cos A
a = b cos C + c cos A

These are useful when solving triangles without using sine or cosine rules directly.

4. Napier’s Analogies

Napier’s analogies are used to avoid the ambiguous case that sometimes occurs in sine rule:

First Analogy:

tan[(A−B)/2] = (a−b) / (a+b) · cot(C/2)

Second Analogy:

tan[(A+B)/2] = (a+b) / (c) · tan(C/2)

These formulas help in finding angles when direct rule application is complicated.

5. Area of a Triangle

(a) Using Two Sides and the Included Angle

Area = (1/2) bc sin A
Area = (1/2) ca sin B
Area = (1/2) ab sin C

(b) Heron’s Formula (Using All 3 Sides)

Let s = (a + b + c) / 2 (semi-perimeter).

Area = √[s(s − a)(s − b)(s − c)]

(c) Using Circumradius

Area = abc / (4R)

6. Right-Angle Triangle Relations

When one angle is 90°, trigonometric ratios simplify:

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

These formulas are used when one angle is given as 90° and one or two sides are known.

Quick Revision Points

  • Sine Rule → Good for AAS, ASA, SSA.
  • Cosine Rule → Good for SSS and SAS.
  • Projection Rule → Sides expressed as sum of projections.
  • Heron’s Formula → Area from three sides.
  • Napier’s Analogies → Used to remove ambiguity in angle calculations.

Detailed Notes:

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