the law of sines: the basics
and angles a, b, and c. draw the
altitude from vertex c to the side
across c; by definition it divides the
original triangle into two right angle
triangles. mark the length of this line h.
it can be observed that:
$\sin a = \frac{h}{b}$ and $ = b\sin a $ and $ h = a\sin b$
$\frac{b\sin a}{ab} = \frac{a\sin b}{ab} \
ightarrow \frac{\sin a}{a} = \frac{\sin b}{b}$
doing the same thing with an altitude from vertex a and side a will yield:
given the triangle $\begin{matrix} & & 23 \\ & & \diagdown & \diagup \\ & 58^\circ & & 27^\circ & & x \\ & & \diagup & \diagdown \\ & & & \end{matrix}$, find the length of side $x$ using the law of sines. round your
final answer to 4 decimal places.
$x = \square$
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Accepted Answer

# Explanation:
## Step1: Find the third angle
The sum of angles in a triangle is $180^{\circ}$. Let the third angle be $C$. Then $C = 180-(58 + 27)=95^{\circ}$
## Step2: Apply the Law of Sines
The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Here, if we assume the side opposite to $58^{\circ}$ is $23$ (let $A = 58^{\circ}, a = 23$) and the side opposite to $27^{\circ}$ is $x$ (let $B=27^{\circ}, b = x$). Then $\frac{x}{\sin27^{\circ}}=\frac{23}{\sin58^{\circ}}$
## Step3: Solve for $x$
$$
\begin{align*}
x&=\frac{23\times\sin27^{\circ}}{\sin58^{\circ}}\
\sin27^{\circ}&\approx0.4540,\sin58^{\circ}\approx0.8480\
x&=\frac{23\times0.4540}{0.8480}\
x&=\frac{10.442}{0.8480}\
x&\approx12.3137
\end{align*}
$$

# Answer:
$12.3137$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the third angle

Step2: Apply the Law of Sines

Step3: Solve for \(x\)

Answer: