To solve Part E and relate [tex]\(a, c, A\)[/tex], and [tex]\(C\)[/tex], we will utilize trigonometric identities and the Law of Sines. Given the relationships and the work from the previous parts, let's proceed step by step.
Step-by-Step Solution:
### 1. Identify Trigonometric Relationships
From the previous parts, we have the following trigonometric identities:
[tex]\[ \sin C = \frac{BD}{a} \][/tex]
[tex]\[ BD = a \sin C \][/tex]
### 2. Apply the Law of Sines
The Law of Sines states that in any triangle:
[tex]\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \][/tex]
Given the identities and the Law of Sines, we can focus on:
[tex]\[ \frac{a}{\sin A} = \frac{c}{\sin C} \][/tex]
### 3. Rewrite as a Ratio Equality
To relate [tex]\(a, c, A\)[/tex], and [tex]\(C\)[/tex], we rewrite the equality above by isolating the terms involving [tex]\(a\)[/tex] and [tex]\(c\)[/tex] on one side and the trigonometric functions on the other side. This yields:
[tex]\[ \frac{a}{c} = \frac{\sin A}{\sin C} \][/tex]
### 4. Final Expression
Thus, the required expression relating [tex]\(a, c, A\)[/tex], and [tex]\(C\)[/tex] can be written as:
[tex]\[ \boxed{\frac{a}{c} = \frac{\sin A}{\sin C}} \][/tex]
This concludes the derivation of the relationship between the sides [tex]\(a, c\)[/tex] and the angles [tex]\(A, C\)[/tex] in the triangle using trigonometric identities and the Law of Sines.
