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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-

image

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

image1

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,

image2

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,

image3

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,

image4

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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