Law of tangents

The Law of Tangents

The Law of Tangents is a mathematical formula that relates the ratios of the lengths of the sides of a triangle to the tangents of the angles of the triangle. The law can be useful in solving triangles when not all of the side lengths are known.

In a triangle ABC, the Law of Tangents states that:

$\frac{a-b}{a+b} = \frac{\tan(\frac{1}{2}(A-B))}{\tan(\frac{1}{2}(A+B))} = \frac{\cot(\frac{1}{2}(C))}{\cot(\frac{1}{2}(A+B))}$

where a, b and c are the lengths of the sides opposite to angles A, B and C respectively.

The Law of Tangents is particularly useful when two sides and the angle opposite to one of the sides are known. In this situation, we can use the formula to solve for the other two angles and the length of the third side.

Let's work through an example to see how the Law of Tangents can be used to solve a triangle:

Example

Suppose we are given a triangle ABC with sides of length a = 5, b = 7 and angle A = 45°. We can use the Law of Tangents to solve for the other two angles and the length of the third side.

Using the Law of Tangents, we have:

$\frac{5-7}{5+7} = \frac{\tan(\frac{1}{2}(45-B))}{\tan(\frac{1}{2}(45+B))} = \frac{\cot(\frac{1}{2}(C))}{\cot(\frac{1}{2}(45+B))}$

Simplifying, we get:

$-\frac{1}{6} = \frac{\tan(\frac{1}{2}(45-B))}{\tan(\frac{1}{2}(45+B))} = \frac{\cot(\frac{1}{2}(C))}{\cot(\frac{1}{2}(45+B))}$

We can solve for B and C using the fact that:

$\tan(\frac{1}{2}(45-B)) = \frac{1-\cos(B)}{\sin(B)} \qquad \cot(\frac{1}{2}(C)) = \frac{\sin(C)}{1+\cos(C)}$

After some algebraic manipulation, we will arrive at:
$B = 15.9^\circ, C = 119.1^\circ$

We can use the Law of Sines to find the length of the third side:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

We can substitute in the values we have found and solve for c:

$\frac{5}{\sin(45^\circ)} = \frac{7}{\sin(15.9^\circ)} = \frac{c}{\sin(119.1^\circ)}$

This gives us:

$c = 9.77$

Therefore, the length of the third side is approximately 9.77.

In conclusion, the Law of Tangents is a useful formula for solving triangles in situations where not all the side lengths are known. By using this formula, we can find missing side lengths and angles, helping us to solve a wide range of problems.

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