3. TRIGONOMETRY: SIDES AND ANGLES OF A TRIANGLE

Basics — Naming the Sides and Angles

In any triangle ABC:

  • Side a is opposite angle A (that is side BC).
  • Side b is opposite angle B (that is side CA).
  • Side c is opposite angle C (that is side AB).

Theorem — Relationship Between Sides and Angles

In any triangle ABC, the sides and angles follow several standard trigonometric formulas. These formulas help in finding unknown angles and sides, especially in non-right-angled triangles.

Cosine Rule

To find the cosine of an angle in terms of the sides:

a² = b² + c² − 2bc cos A

Similarly,

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

These are the Cosine Rule. Use them when all three sides are known or when two sides and included angle are known.

Sine Rule

To express the sine of an angle in terms of the sides:

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

Here, R is the circumradius of the triangle.

Half-Angle Formulas

1. Sine of Half-Angles

To find the sine of half-angles in terms of sides:

sin(A/2) = √[(s − b)(s − c) / (bc)]

Similarly, formulas exist for B/2 and C/2.

Here, s is the semi-perimeter:

s = (a + b + c) / 2

2. Cosine of Half-Angles

cos(A/2) = √[s(s − a) / (bc)]

3. Tangent of Half-Angles

tan(A/2) = √[(s − b)(s − c) / (s(s − a))]

Formula to Express Sine of Any Angle in Terms of the Sides

Using the standard trigonometric relationships, we can express sin A in terms of the sides of a triangle. These relationships help in solving triangles without needing heights or additional geometric constructions.

Examples

The PDF provides multiple worked examples demonstrating use of:

  • Cosine rule
  • Sine rule
  • Half-angle formulas
  • Side–angle relationships

Examples generally follow this pattern:

  1. Identify known sides or angles.
  2. Choose appropriate rule (sine, cosine, or half-angle formula).
  3. Substitute values and simplify to find missing side or angle.

Practice Problems

The PDF includes several “Practice Examples” for students to strengthen problem-solving skills. They involve calculating unknown sides or angles using the above formulas.

Quick Revision Points

  • a, b, c are sides opposite A, B, C.
  • Cosine Rule relates sides with the cosine of included angles.
  • Sine Rule gives proportional relationship between sides and angles.
  • Half-Angle Formulas useful for advanced trigonometric calculations.
  • Know semi-perimeter s = (a + b + c)/2.

Detailed Notes:

For PDF style full-color notes, open the complete study material below:

Textbook Link

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