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\\)
\\(\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 \\(\triangle\\) with angles \\(47^\circ\\), \\(19^\circ\\) and side 30, find the length of side \\(x\\) using the law of sines. round your final answer to 4 decimal places.
\\(x = \square\\)
question help: video

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180^{\circ}-47^{\circ}-19^{\circ}=114^{\circ}\).

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, let \(a = 30\), \(A = 114^{\circ}\), \(B=19^{\circ}\), and \(b = x\). So \(\frac{30}{\sin114^{\circ}}=\frac{x}{\sin19^{\circ}}\).
Cross - multiply to get \(x=\frac{30\times\sin19^{\circ}}{\sin114^{\circ}}\).
We know that \(\sin114^{\circ}=\sin(90^{\circ} + 24^{\circ})=\cos24^{\circ}\approx0.9135\) and \(\sin19^{\circ}\approx0.3256\).
Then \(x=\frac{30\times0.3256}{0.9135}=\frac{9.768}{0.9135}\approx10.6929\).

Answer:

\(10.6929\)