Law Of Tangents

The laws of tangent (Law of Tan) describes the relation between difference and sum of sides of a right triangle and tangents of half of the difference and sum of corresponding angles. It represents the relationship between the tangent of two angles of a triangle and the length of the opposite sides. The law of tangents is also applied to a non-right triangle and it is equally as powerful like the law of sines and the law of cosines. It can be used to find the remaining parts of a triangle if two angles and one side or two sides and one angle are given which are referred to as side-angle-side(SAS) and angle-side-angle(ASA), from the congruence of triangles concept.

To understand the law of tangents in a better way, you need some pieces of information for a general triangle even it may or may not be a triangle. The four cases involved are:

• Two sides and one opposite angle
• One side and two angles
• Three Sides
• Two sides and the angle between them

Formulas For Laws Of Tangents

Let us assume a right triangle ABC in which sides opposite to

Similarly for other sides,

Since tan (-θ)= -tan θ for any angle θ, we can switch the order of letters in the above law of tangents formulas and can be rewritten as:

Similarly for other sides,

The formulas (1), (2), and (3) are used when a>b, b>c, and c>a, and the formulas (4), (5) and (6) are used when b>a, c>b and a>c.

Laws Of Tangent Proof

To Prove:

Proof:From the law of Sine,

Use first and second relation,

a = k sin A and b = k sin B

From this,

a – b = k (sin A – sin B)

a + b = k (sin A + sin B)

So, we get

Identity Formulas for Sine are:

Substitute those formulas in equation (1),we get

Hence Proved.

Practice problem

Question :

Solve the triangle

Solution :

We know that,

∠A + ∠B + ∠C = 180°

∠A + ∠B=  180°- ∠C  = 180° – 96° = 84°

By law of tangents,

for a triangle ABC with sides a, b and c respective to the angles A , B and C is given by,

Therefore,

A – B = 25.4°

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