Question

use the law of sines to find the indicated side length in this triangle. 5 120°? 23° round your answer to the nearest hundredth.

Explanation:

Step 1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Let the third angle be \(A\). So, \(A = 180^\circ - 120^\circ - 23^\circ = 37^\circ\).

Step 2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let the side opposite \(120^\circ\) be \(a\), the side opposite \(23^\circ\) be \(b\), and the side opposite \(37^\circ\) be \(5\) (wait, no: the side given is \(5\), let's correct. Let the side with length \(5\) be opposite angle \(23^\circ\)? Wait, no, looking at the triangle: the angle \(120^\circ\), angle \(23^\circ\), and the side of length \(5\) – let's define: let’s say the side of length \(5\) is opposite angle \(37^\circ\)? Wait, no, let's re - define. Let’s denote: angle \(B = 120^\circ\), angle \(C=23^\circ\), so angle \(A = 37^\circ\). Let the side opposite angle \(A\) ( \(37^\circ\)) be \(a = 5\), and the side opposite angle \(B\) ( \(120^\circ\)) be \(x\) (the one we need to find). Then by Law of Sines: \(\frac{x}{\sin B}=\frac{a}{\sin A}\). So \(x=\frac{a\sin B}{\sin A}\). Substituting \(a = 5\), \(B = 120^\circ\), \(A=37^\circ\). \(\sin120^\circ=\frac{\sqrt{3}}{2}\approx0.8660\), \(\sin37^\circ\approx0.6018\). Then \(x=\frac{5\times\sin120^\circ}{\sin37^\circ}=\frac{5\times0.8660}{0.6018}\approx\frac{4.33}{0.6018}\approx7.19\) (Wait, maybe I mixed up the angles. Wait, maybe the side of length \(5\) is opposite angle \(23^\circ\). Let's re - check. Let's say: angle \(C = 23^\circ\), side \(c = 5\), angle \(B=120^\circ\), angle \(A = 37^\circ\), and we need to find side \(b\) (opposite angle \(B\)). Then by Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\). So \(b=\frac{c\sin B}{\sin C}\). Substituting \(c = 5\), \(\sin B=\sin120^\circ\approx0.8660\), \(\sin C=\sin23^\circ\approx0.3907\). Then \(b=\frac{5\times0.8660}{0.3907}=\frac{4.33}{0.3907}\approx11.08\)? Wait, no, I think I messed up the angle - side correspondence. Let's start over.

The triangle has angles: \(120^\circ\), \(23^\circ\), and \(180-(120 + 23)=37^\circ\). Let's assume that the side of length \(5\) is opposite the \(37^\circ\) angle. Then the side we want to find (let's call it \(x\)) is opposite the \(120^\circ\) angle. Then by Law of Sines: \(\frac{x}{\sin120^\circ}=\frac{5}{\sin37^\circ}\). So \(x=\frac{5\times\sin120^\circ}{\sin37^\circ}\). \(\sin120^\circ=\sin(60^\circ)=\frac{\sqrt{3}}{2}\approx0.8660\), \(\sin37^\circ\approx0.6018\). Then \(x=\frac{5\times0.8660}{0.6018}=\frac{4.33}{0.6018}\approx7.19\)? Wait, no, maybe the side of length \(5\) is opposite the \(23^\circ\) angle. Let's try that. If the side of length \(5\) is opposite \(23^\circ\), and we want to find the side opposite \(120^\circ\) (let's call it \(x\)). Then \(\frac{x}{\sin120^\circ}=\frac{5}{\sin23^\circ}\). \(\sin23^\circ\approx0.3907\), \(\sin120^\circ\approx0.8660\). Then \(x=\frac{5\times0.8660}{0.3907}=\frac{4.33}{0.3907}\approx11.08\). Wait, now I see my mistake. Let's find the correct angle - side correspondence. The angle of \(120^\circ\) is an obtuse angle, so the side opposite to it should be the longest side. The side of length \(5\): let's check the angles. The angle \(23^\circ\) is acute, \(37^\circ\) is acute, \(120^\circ\) is obtuse. So the side opposite \(23^\circ\) is the shortest side, the side opposite \(37^\circ\) is longer, and the side opposite \(120^\circ\) is the longest. So if the side of length \(5\) is opposite \(37^\circ\), then the side opposite \(120^\circ\) is longer than \(5\), and the side opposite \(23^\circ\) is shorter than \(5\). Let's confirm with the Law of Sines. Let’s define:

Let \(a\) be the side opposite angle \(A = 37^\circ\), so \(a = 5\)

Let \(b\) be the side opposite angle \(B=120^\circ\)

Let \(c\) be the side opposite angle \(C = 23^\circ\)

By Law of Sines: \(\frac{b}{\sin B}=\frac{a}{\sin A}\)

So \(b=\frac{a\sin B}{\sin A}=\frac{5\times\sin120^\circ}{\sin37^\circ}\)

\(\sin120^\circ=\sin(180 - 60)=\sin60=\frac{\sqrt{3}}{2}\approx0.8660\)

\(\sin37^\circ\approx0.6018\)

\(b=\frac{5\times0.8660}{0.6018}=\frac{4.33}{0.6018}\approx7.19\)? Wait, that can't be, because \(120^\circ\) is a large angle, the side opposite should be longer. Wait, I think I mixed up the angle for the side of length \(5\). Maybe the side of length \(5\) is opposite \(23^\circ\). Let's try that.

Let \(c = 5\) (opposite angle \(C = 23^\circ\))

\(b\) is opposite angle \(B = 120^\circ\)

Then by Law of Sines: \(\frac{b}{\sin B}=\frac{c}{\sin C}\)

\(b=\frac{c\sin B}{\sin C}=\frac{5\times\sin120^\circ}{\sin23^\circ}\)

\(\sin23^\circ\approx0.3907\)

\(b=\frac{5\times0.8660}{0.3907}=\frac{4.33}{0.3907}\approx11.08\)

Now, let's check the sum of angles again: \(120 + 23+37 = 180\), correct. Now, let's see the lengths: side opposite \(23^\circ\) is \(5\), side opposite \(37^\circ\) would be \(\frac{5\times\sin37^\circ}{\sin23^\circ}=\frac{5\times0.6018}{0.3907}\approx7.73\), and side opposite \(120^\circ\) is \(11.08\). Since \(120^\circ\) is the largest angle, its opposite side should be the longest, which \(11.08\) is longer than \(7.73\) and \(5\), so this makes sense. I made a mistake earlier in assigning the side of length \(5\) to the wrong angle. So the correct assignment is: the side of length \(5\) is opposite \(23^\circ\), and we need to find the side opposite \(120^\circ\).

So, \(b=\frac{5\times\sin120^\circ}{\sin23^\circ}\)

\(\sin120^\circ=\frac{\sqrt{3}}{2}\approx0.8660\)

\(\sin23^\circ\approx0.3907\)

\(b=\frac{5\times0.8660}{0.3907}=\frac{4.33}{0.3907}\approx11.08\)

Answer:

\(11.08\)