Answer:
1.25 < y <= 8.32124
Step-by-step explanation:
We use the law of cosines for two triangles:
[tex]c^2 = a^2 + b^2 - 2 a b \cos(\gamma) \\ d^2 = a^2 + b^2 - 2 a b \cos(\delta)[/tex]
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 [tex]c,\, d,\, \gamma,\, \text{and}\ \delta[/tex] 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 a and b, but different values for c and γ . Here I use the constants [tex]c = 6\,\ \gamma = 43°\, \ d = 4[/tex]. The following equations and inequalities are supplied to Reduce, with an Extential quantifier specifying that Reduce should discover the range of values for [tex]\cos(\gamma)[/tex].
[tex]\text{problem}=\exists _{\{a,b\}}\left(\begin{array}{ccc}36=a^2+b^2-2 a b \cos (43 {}^{\circ})\ \ \land\\16=a^2+b^2-2 a b \cos (\delta )\,\,\land\\ 0<a\land 0<b\land -1<\cos (\delta )<1\end{array}\right)\\ \\ \text{variables}=\{a,b,\cos (\delta )\}\\ \\ \text{red}=\text{Reduce}[\text{problem},\text{variables},\mathbb{R}]\,\,\,\,\,\text{gives}\,\,\,\,\,\frac{1}{9} (4 \cos (43 {}^{\circ})+5)\leq \cos (\delta )<1\\\\\text{N}[\text{red}]\,\,\,\text{gives}\,\,\,0.880602\leq \cos (\delta )<1.[/tex]
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.
[tex]\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}<y\leq \frac{1}{4} \left(\frac{\cos ^{-1}}{{{}^{\circ}}}\left(\frac{1}{9} (4 \cos (43 {}^{\circ})+5)\right)}+5\right)[/tex]
