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[tex]\[
\begin{aligned}
\sin C & = \frac{BD}{a} \\
BD & = a \sin C
\end{aligned}
\][/tex]

Part E:
Combine the two expressions found in parts C and D to relate [tex]\( a, c, A \)[/tex], and [tex]\( C \)[/tex]. Rewrite the expression as an equality between two ratios. Show your work.


Sagot :

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.
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