Law of Tangents

The Law of Tangents links the difference and sum of two sides of a triangle to half the difference and half the sum of their opposite angles. It comes from rewriting the sine rule and using standard tangent identities, so it is especially useful when you know two sides and the two opposite angles, or when you want to eliminate an unknown side ratio. Because it uses half-sums and half-differences, it is often a good companion to the Law of Sines in oblique triangles.

Use it in non-right triangles when you know or can form the pairs (a, A) and (b, B), especially if a difference between the sides is easier to work with than the sides themselves. It is common in advanced triangle-solving, surveying, and trigonometric proof problems.

• Forgetting that the formula uses half the sum and half the difference of the angles, not the full angles
• Mixing up which side matches which angle; each side must be paired with its opposite angle
• Using the formula in a right triangle without checking that the angle pairs make sense for an oblique triangle