Sine and Cosine Rule Equations

Basic Rule

• a = b = c
  sin A sin B sin C
• sin A = sin B = sin C
  a b c

Finding the length of a side

• a = b
  sin A sin B
• a = c
  sin A sin C
• b = c
  sin B sin C

Finding an angle

• sin A = sin B
  a b
• sin A = sin C
  a c
• sin B = sin C
  b c

Two Solutions

• when the angle you are finding is larger than the given angle
• sin (180-x) = sin x

Basic Rule

• a^2 = b^2 + c^2 - 2bc cos A
• cos A = b^2 + c^2 - a^2
  2bc

Finding an unknown side

• a^2 = b^2 + c^2 - 2bc cos A
• b^2 = a^2 + c^2 - 2bc cosB
• c^2 = a^2 + b^2 - 2bc cosC

Finding an unknown angle

• cos A = b^2 + c^2 - a^2
  2bc
• cos B = a^2 + c^2 - b^2
  2bc
• cos C = a^2 + b^2 - c^2
  2bc

Right- angled or isosceles

• sine theorem (sin A = a/b)
• cosine theorem (cos A = c/b)
• tangent theorem (tan A = a/c)
• Pythagoras' theorem (a^2 + b^2 = c^2)

Two sides and the included angle

• cosine rule

Other combinations

• sine rule

Basic Rule

• 1/2 ab sin C
• 1/2 ac sin B
• 1/2 bc sin A