Associated Topics || Dr. Math Home || Search Dr. Math

### Law of Tangents

```
Date: 01/22/99 at 01:23:14
From: Matthew Veatch
Subject: Law of tangents

For my pre-calc class, I am trying to prove and derive the law of
```

```
Date: 01/22/99 at 12:44:51
From: Doctor Floor
Subject: Re: Law of tangents

Hi Matthew,

Thank you for sending your question to Dr. Math!

Let a and b be two sides of a triangle. Let A be the angle opposite a
and B be the angle opposite b. The law of tangents says:

a+b   tan(0.5(A+B))
--- = -------------
a-b   tan(0.5(A-B))

To derive this formula, first consider the sum and subtraction formulas
for sines:

sin(t+u) = sin(t)cos(u) + cos(t)sin(u)
sin(t-u) = sin(t)cos(u) - cos(t)sin(u)

Adding and subtracting these two gives:

sin(t+u) + sin(t-u) = 2sin(t)cos(u)
sin(t+u) - sin(t-u) = 2cos(t)sin(u)

Now let t = 0.5(A+B) and u = 0.5(A-B), then t+u = A and t-u = B.
This gives:

sin(A) + sin(B) = 2sin(0.5(A+B))cos(0.5(A-B))
sin(A) - sin(B) = 2sin(0.5(A-B))cos(0.5(A+B))

So we can derive the following:

tan(0.5(A+B))   sin(0.5(A+B))cos(0.5(A-B))
------------- = --------------------------
tan(0.5(A-B))   sin(0.5(A-B))cos(0.5(A+B))

2sin(0.5(A+B))cos(0.5(A-B))
= ---------------------------
2sin(0.5(A-B))cos(0.5(A+B))

sin(A) + sin(B)
= ---------------
sin(A) - sin(B)

And using the law of sines this equals:

a + b
=  -----
a - b

as desired!

If you have a math question again, please send it to Dr. Math.

Best regards,

- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search