Relation between sides and angles of a triangle

Sine Rule:

The sides of a triangle are proportional to the sines of the angles opposite to them.

a / sin A = b / sin B = c / sin C = 2R (Or) sin A /a = sin B /b = sin C/c = 1/2R

Where R is the circum radius of the circle.

Ex:

If R = 3, a = 6 Then ∟A =?

Sol:

a = 2Rsin A => 6 = 6 sin A

=> sin A = 1 => ∟A = 90⁰

Cosine Rule:

The  angle of a triangle are related to three sides of a triangle by cosine rule.

cos A = b² + c² – a² / 2bc (or) a² = b² + c² – 2bc cos A

cos B = a² + c² - b² / 2ac (or) b² = c² + a² – 2ac cos B

cos C = a² + b² - c² / 2ab (or) c² = a² + b² – 2ab cos C

Ex:

If a² + b² = c² find cos C

Sol:

cos C = a² + b² - c² / 2ab = 0/ 2ab = 0 => ∟C = 90⁰