[Math] Find $k$ such that the angle between the vectors $(2,k)$ and $ (3,5)$ is $60$ degrees

Question

I have $2$ vectors : $U =(2,k)$ and $V = (3,5)$. I want to find the $k$ value when the angle between $U$ and $V$ is $60$ degrees.

This what I tried to do but I don't get the right answer :

$2\cdot3 + 5k = \sqrt{4+k^2} \cdot \sqrt{34} \cdot \cos60 \rightarrow 24k^2 +30.84k+5$

Answer 1

The angle between V and the X axis is $arctan(5/3) = 59 \deg.$

Therefore, if $U$ is about 1 deg. below the X axis, the angle between U and V will be 60 degrees.

Now solve:

$k/2 = tan(-1)$

and get the solution.