Answer:
1.25 < y <= 8.32124
Step-by-step explanation:
We use the law of cosines for two triangles:

This answer shows how to use the Reduce and Exists functions of Mathematica to solve either this problem, or the general problem of a pair of triangles with two sides of one triangle equal to two sides of the other triangle. That answer (with
as independent variables which must be given ranges) has 11 cases, and would be a terrible waste of time to find by hand.
The law of cosines is used twice, with the same values for <em>a</em> and <em>b</em>, but different values for <em>c</em> and <em>γ</em> . Here I use the constants
. The following equations and inequalities are supplied to Reduce, with an Extential quantifier specifying that Reduce should discover the range of values for
.

This proves (since we used Reduce, not Solve, which is less reliable) that
- a triangle exists that has angle 43°, two adjacent sides of length a and b and opposite side of length 6, and that
- a second triangle exists with unknown angle, adjacent sides a and b equal to the corresponding sides of the first triangle, and opposite side length 4.
- There is only one range of angles which satisfy the requirements.
![\text{mincos}=\text{First}[\text{red}]\ \ \text{gives}\ \ \frac{1}{9} (4 \cos (43 {}^{\circ})+5)\\\text{maxcos}=\text{Last}[\text{red}]\ \ \text{gives}\ \ 1\\\\\text{Solve}[4 y-5=\delta,y ]\ \ \ \text{gives}\ \ \ \left\{\left\{y\to \frac{\delta }{4}+\frac{5}{4}\right\}\right\}\\ \\\\\ \frac{5}{4}](https://tex.z-dn.net/?f=%5Ctext%7Bmincos%7D%3D%5Ctext%7BFirst%7D%5B%5Ctext%7Bred%7D%5D%5C%20%5C%20%5Ctext%7Bgives%7D%5C%20%5C%20%5Cfrac%7B1%7D%7B9%7D%20%284%20%5Ccos%20%2843%20%7B%7D%5E%7B%5Ccirc%7D%29%2B5%29%5C%5C%5Ctext%7Bmaxcos%7D%3D%5Ctext%7BLast%7D%5B%5Ctext%7Bred%7D%5D%5C%20%5C%20%5Ctext%7Bgives%7D%5C%20%5C%201%5C%5C%5C%5C%5Ctext%7BSolve%7D%5B4%20y-5%3D%5Cdelta%2Cy%20%5D%5C%20%5C%20%5C%20%5Ctext%7Bgives%7D%5C%20%5C%20%5C%20%5Cleft%5C%7B%5Cleft%5C%7By%5Cto%20%5Cfrac%7B%5Cdelta%20%7D%7B4%7D%2B%5Cfrac%7B5%7D%7B4%7D%5Cright%5C%7D%5Cright%5C%7D%5C%5C%20%5C%5C%5C%5C%5C%20%5Cfrac%7B5%7D%7B4%7D%3Cy%5Cleq%20%5Cfrac%7B1%7D%7B4%7D%20%5Cleft%28%5Cfrac%7B%5Ccos%20%5E%7B-1%7D%7D%7B%7B%7B%7D%5E%7B%5Ccirc%7D%7D%7D%5Cleft%28%5Cfrac%7B1%7D%7B9%7D%20%284%20%5Ccos%20%20%20%20%2843%20%7B%7D%5E%7B%5Ccirc%7D%29%2B5%29%5Cright%29%7D%2B5%5Cright%29)