Question

How do you find the exact value of `arctan(1/2)+arctan(1/3)`?

Answer 1

Suppose `alpha = arctan(1/2)` and `beta = arctan(1/3)`

What is the value of `tan(alpha + beta)` ?

It's probably easiest to calculate `sin alpha`, `cos alpha`, `sin beta` and `cos beta` first:

If `alpha = arctan(1/2)`, then `1/2 = tan alpha = sin alpha / cos alpha`.

Multiplying through by `2cos alpha` we get

`cos alpha = 2 sin alpha`

Squaring both sides and using `sin^2 alpha + cos^2 alpha = 1` we get

`4 sin^2 alpha = cos^2 alpha = 1 - sin^2 alpha`

Adding `sin^2 alpha` to both sides and dividing by 5 we get

`sin^2 alpha = 1/5`

So `sin alpha = 1/sqrt(5)` and `cos alpha = 2 sin alpha = 2/sqrt(5)`.

Similarly, we can find `sin beta = 1/sqrt(10)` and `cos beta = 3/sqrt(10)`.

Now we can calculate

`tan(alpha + beta) = sin(alpha+beta)/cos(alpha+beta)`

`=(sin alpha cos beta + sin beta cos alpha)/(cos alpha cos beta - sin alpha sin beta)`

The numerator:

`sin alpha cos beta + sin beta cos alpha`

`=(1/sqrt(5))(3/sqrt(10)) + (1/sqrt(10))(2/sqrt(5))`

`=5/sqrt(50) = 1/sqrt(2)`

The denominator:

`cos alpha cos beta - sin alpha sin beta`

`(2/sqrt(5))(3/sqrt(10)) - (1/sqrt(5))(1/sqrt(10))`

`=5/sqrt(50) = 1/sqrt(2)`

So putting these together:

`tan(alpha + beta) = (1/sqrt(2))/(1/sqrt(2)) = 1`

So `alpha + beta = pi/4`