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# Laws‌ ‌of‌ ‌Tangent‌

Laws of tangent represent a relationship between the sides of a right-angled triangle and the tangents of its angles. It is similar to the law of sines and cosines but with a bit of variation. It is used to find the other sides of a triangle if its sides are known. It works for every triangle and is not restricted to only right-angled triangles.

## Statement of the law of tangent

According to the law of tangents, the ratio of sum and difference of any two sides of a triangle is equal to the tangent ratio of half the sum and tangent of half the difference of the angles opposite to the corresponding sides.

## Formula from the law of tangent

To calculate the formula from the tangent law, let us consider a right triangle ABC whose sides opposite to ∠A, ∠B, and ∠C are a, b and c, respectively. According to the laws of a tangent, we can get the following relations-

The first three formulas are used when a>b, b>c or c>a. Whereas, the other three formulas are used when b>a, c>b, a>c.

## Proof

a = k sin A and b = k sin B
Also, we will get,
a – b = k (sin A – sin B)
a + b = k (sin A + sin B)
Therefore, the equation becomes,

We know the identities of sine-

sin A – sin B = 2 cos (A+B /2) sin (A-B /2)
sin A + sin B = 2 sin (A+B /2) cos (A-B /2)

Substituting and rearranging the above equation, we get,

Hence, proved.

## Example

For the above triangle, solve when a = 5, b = 3, angle C = 96 degrees. Also, find the value of A-B.

## Solution

We know the sum of angles inside a triangle is equal to 180 degrees.

∠A + ∠B + ∠C = 180°

∠A + ∠B = 180°- ∠C = 180° – 96° = 84°

From the law of tangents, the formula for a triangle ABC with sides a, b and c respectively to the angles A, B and C is written as,

Also, ½ (A-B) = 12.7

We get, A-B = 25.4 degrees.

## Fun Facts about the law of tangent

1. The laws of tangent were firstly used for spherical triangles.
2. A Persian mathematician, Nasir al-Din al-Tusi, discovered them, who proved the law of sines for plane triangles.
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