Question

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\\)
\\(b\sin a = a\sin b\\) \\(\
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 \\(\

$$\begin{matrix} & & 23 \\\\ & & \\\\ & 14^\\circ & 23^\\circ & x \\\\ \\end{matrix}$$

\\) , find the length of side \\(x\\) using the law of sines. round your
final answer to 4 decimal places.
\\(x = \square\\)
question help: \\(\boldsymbol{\text{video}}\\)

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180^{\circ}-(14^{\circ}+23^{\circ})=143^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a = 23\), \(A = 143^{\circ}\), \(B=23^{\circ}\), and \(b = x\). Then \(\frac{23}{\sin143^{\circ}}=\frac{x}{\sin23^{\circ}}\).
Since \(\sin143^{\circ}=\sin(180 - 37)^{\circ}=\sin37^{\circ}\approx0.6018\) and \(\sin23^{\circ}\approx0.3907\).
We can solve for \(x\): \(x=\frac{23\times\sin23^{\circ}}{\sin143^{\circ}}\).
Substitute the values: \(x=\frac{23\times0.3907}{0.6018}\).
Calculate \(23\times0.3907 = 8.9861\).
Then \(x=\frac{8.9861}{0.6018}\approx14.9320\).

Answer:

\(14.9320\)