To solve for the unknown side or angle in triangle [tex]\( \triangle ABC \)[/tex] with given information [tex]\( m\angle A = 35^\circ \)[/tex], [tex]\( m\angle B = 40^\circ \)[/tex], and [tex]\( a = 9 \)[/tex], it is important to understand which equations are relevant for trigonometric computations in a triangle.
Given the task, it appears there might be some typographical errors or missing context in the question. Nevertheless, I will interpret the question to mean which trigonometric rule or formula should be used for finding the unknown side in relation to the given angle measures and the side length.
First, remind ourselves of two important trigonometric laws that can be used in solving for sides and angles in a triangle:
1. Law of Sines states:
[tex]\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\][/tex]
For [tex]\(\triangle ABC\)[/tex], if we know [tex]\(a\)[/tex], [tex]\(\angle A\)[/tex], and [tex]\(\angle B\)[/tex], we can find side [tex]\(b\)[/tex] (opposite to [tex]\(\angle B\)[/tex]) using:
[tex]\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\][/tex]
Thus:
[tex]\[
b = a \cdot \frac{\sin B}{\sin A}
\][/tex]
2. Law of Cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cdot \cos C
\][/tex]
This law allows computing the third side or an included angle when two sides and the included angle are known.
Given the provided information ([tex]\(a = 9\)[/tex], [tex]\(m\angle A = 35^\circ\)[/tex] and [tex]\(m\angle B = 40^\circ\)[/tex]), we should first confirm that those two angles sum to less than [tex]\(180^\circ\)[/tex]:
[tex]\[
m\angle C = 180^\circ - m\angle A - m\angle B = 180^\circ - 35^\circ - 40^\circ = 105^\circ
\][/tex]
Now, with angle measures known, we can use the Law of Sines to find side [tex]\(b\)[/tex].
Thus, the appropriate equation from these choices, interpreting and correcting as needed, is:
B. [tex]\(\sin y = \sin g\)[/tex]
After verification, using the Law of Sines is indeed the solution which interprets the expression related correctly for solving the side lengths. Therefore, option B (interpreted as using the Law of Sines) would be the right choice to find the unknown side length.
