# The Law of Tangents | Proof & How it works?

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It is a relationship between the tangents of two angles and two sides in a triangle. You can easily find out the any one missing variable (value or angle) if the corresponding sides and angles are known.

This is the formula for the Law of Tangents, looks tough but it is pretty easier formula, you will know more about it by scrolling down.

Well, you don’t need to worry about the three different formula, it is how you can use a single formula for three different sides of a triangle.

where a, b, and c are sides and A, B, and C are their corresponding angles.

This law states that the difference and summation of sides divided by each other equals to the half the tangent of difference and summation of sides divided by each other.

Let’s try out a problem based on this law,

Q1: Solve the triangle ABC using Law of Tangents, given a = 48, b =32, and C = 57°.

Solution: α – β = 180° – γ
= 180° – 57°
= 61.5°

Also, α + β = 123° –(2)

Using the Law of Tangents formula,

α – β = 61.5° → (1)
α + β = 123° → (2)

Using elimination (adding or subtracting) method on equation (1) and (2),

2α = 163.4°
α = 81.7°

Putting the value of “A” in equation (1) or (2), we get:

81.7° + β = 123°
β = 123° – 81.7°
= 41.3°

Now, you can also use Law of Tangents again to find “c” side but we for our convenience, we use Law of Cosines here.

c² = a² + b² – 2abcosγ
= (48)² +(32)² – 2(48)(32)(cos57°)
= 2304 + 1024 – (3072)(0.54)
= 3328 – 1673.13
= √1654.8
c = 40.68

Q2: Use the Law of Tangents to solve the triangle ABC where a = 925, c = 432, and B = 42°30′.

Solution: α + γ = 180° – β
= 137°30′
》 1/2 (α + γ) = 68°45′
(α – γ) = 86°6′ (2)
》 1/2 (α – γ) = 43°

Using the Law of Tangents formula as stated below:

Inputting the values in the formula, we get:

Thus, (α + γ) = 137°30′ → (1)
(α + γ) = 86°6′ → (2)

Using the Elimination (Addition and Subtraction) method on equation (1) and (2), we get,

2α = 223°36′
α = 111°48′

Also, 2γ = 51°24′
γ = 25°42′

To find the side “b”, you can use either Law of Sines or Law of Cosines, for our convenience, we use Law of Sines.

= 673

Theorem: In any triangle ABC, show that:

Proof: By the Law of Sines in any triangle ABC,

we have,

a = D sin α and b = sin β
a + b = D(sin α + sin β) → (1)
a – b = D(sin α – sin β) → (2)

From (1) and (2), we get:

Using the formulae,

we get,

which finally yields the formula mentioned below of the Law of Tangents,

These three relations are known as the Law of Tangents.

Note: The interchange of lengths a and b results in the interchange of angles A and B. Hence, if b > a, then it is better to use the tangent formula in the form.