Non Right Angled Trig

The sine rule, cosine rule and area formula, each proved from the same triangle. Drag the apex so the height falls inside or outside the triangle and watch the proofs still work. See the Trigonometry Graphs and Trigonometric Proofs pages too. Built for KS4 and KS5.

Each side faces the angle of the same letter: side is opposite angle .

Sine rule

Best for: a matching angle-and-opposite-side pair plus one more part.

Proof Hints
  1. Two equations for the height, using and .
  2. Both equal the same height, so set them equal.
  3. Angles on top → finding angles. Sides on top → finding sides.
Proof

Drop height from , meeting at .

Triangle :

Triangle :

Both equal : . A height from another vertex adds the third ratio.

When the height falls outside

If is obtuse the foot lands outside , so the height leans on the extended base. The right triangle now uses the outside angle , but , so the height is still and the rule is unchanged.

When to use it

The arrows link each angle to the side opposite it. Match a known angle to its opposite side, then use a second pair to reach the missing part. Knowns in green, unknown in red.

The ambiguous case (two possible triangles)

Given two sides and an angle not between them, the side opposite the known angle can swing to two positions, giving two triangles. Drag the slider: when is longer than the height it crosses the base twice.

The sine rule returns the acute angle; the obtuse triangle uses . This is exactly reading two solutions off the sine graph - see Solutions from graphs on the Trigonometry Graphs page.