Question

question 2 of 10
in the triangle below, ( b = )____. if necessary, round your answer to two decimal places.
answer here

Explanation:

Step1: Find angle at B

First, we know that the sum of angles in a triangle is \(180^\circ\). So, angle at \(B = 180^\circ - 34.5^\circ - 29.8^\circ\)
\(= 180 - 34.5 - 29.8 = 115.7^\circ\)

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, we know side \(AC\) is not, wait, the side given is \(BC = 22\) (opposite angle \(A = 34.5^\circ\)), and we need to find \(b\) (which is \(AC\)? Wait, no, in the triangle, side \(b\) is opposite angle \(B\)? Wait, no, let's clarify the notation. In triangle notation, side \(a\) is opposite angle \(A\), side \(b\) opposite angle \(B\), side \(c\) opposite angle \(C\). Wait, the diagram: angle at \(A\) is \(34.5^\circ\), angle at \(C\) is \(29.8^\circ\), side \(BC = 22\) (which is side \(a\), opposite angle \(A\)), and we need to find side \(b\) (opposite angle \(B\)). Wait, no, maybe I mixed up. Wait, the side labeled \(b\) is between \(A\) and \(C\)? Wait, no, standard notation: in triangle \(ABC\), side \(a\) is \(BC\), side \(b\) is \(AC\), side \(c\) is \(AB\). So angle at \(A\) is \(34.5^\circ\), angle at \(C\) is \(29.8^\circ\), side \(a = BC = 22\), and we need to find side \(b = AC\). Wait, no, let's re-express.

Wait, Law of Sines: \(\frac{b}{\sin B}=\frac{BC}{\sin A}\). So \(b=\frac{BC \times \sin B}{\sin A}\)

We have \(BC = 22\), angle \(A = 34.5^\circ\), angle \(B = 115.7^\circ\)

So \(\sin A=\sin(34.5^\circ)\approx0.5664\), \(\sin B=\sin(115.7^\circ)=\sin(180 - 64.3^\circ)=\sin(64.3^\circ)\approx0.9017\)

Then \(b=\frac{22 \times 0.9017}{0.5664}\)

Step3: Calculate the value

First, calculate numerator: \(22 \times 0.9017 = 19.8374\)

Then divide by \(0.5664\): \(19.8374 \div 0.5664 \approx 35.02\)

Wait, wait, maybe I messed up the sides. Wait, maybe the side given is \(BC = 22\) (side \(a\), opposite angle \(A\)), and we need to find side \(b\) (opposite angle \(B\)). Wait, angle \(B = 180 - 34.5 - 29.8 = 115.7^\circ\), correct. Then Law of Sines: \(\frac{b}{\sin B}=\frac{a}{\sin A}\), so \(b = \frac{a \times \sin B}{\sin A}\), where \(a = 22\), angle \(A = 34.5^\circ\), angle \(B = 115.7^\circ\)

So \(\sin(34.5^\circ)\approx0.5664\), \(\sin(115.7^\circ)\approx\sin(115.7)\approx0.901\)

Then \(b = \frac{22 \times 0.901}{0.5664} \approx \frac{19.822}{0.5664} \approx 35.0\) (wait, maybe my calculation of \(\sin(115.7^\circ)\) is off. Let's use more precise values.

\(\sin(34.5^\circ)\): 34.5 degrees. \(\sin(34.5) = \sin(34 + 0.5) = \sin(34^\circ) \cos(0.5^\circ) + \cos(34^\circ) \sin(0.5^\circ) \approx 0.5592 \times 0.99996 + 0.8290 \times 0.008727 \approx 0.5592 + 0.00723 \approx 0.5664\) (correct)

\(\sin(115.7^\circ)\): 115.7 degrees. \(115.7 - 90 = 25.7\), so \(\sin(115.7^\circ) = \cos(25.7^\circ) \approx 0.901\) (more precisely, \(\cos(25.7^\circ) = \cos(25 + 0.7) = \cos(25^\circ)\cos(0.7^\circ) - \sin(25^\circ)\sin(0.7^\circ) \approx 0.9063 \times 0.9999 + 0.4226 \times 0.0122 \approx 0.9063 - 0.00516 \approx 0.9011\))

So \(b = \frac{22 \times 0.9011}{0.5664} = \frac{19.8242}{0.5664} \approx 35.00\) (rounded to two decimal places)

Wait, but let's check again. Wait, maybe the side labeled \(b\) is opposite angle \(B\), and the side given is \(BC = 22\) (opposite angle \(A\)). So yes, Law of Sines: \(\frac{b}{\sin B} = \frac{22}{\sin A}\), so \(b = \frac{22 \times \sin B}{\sin A}\)

Angle \(B = 180 - 34.5 - 29.8 = 115.7^\circ\), correct.

So \(\sin(115.7^\circ) \approx \sin(115.7) \approx 0.901\), \(\sin(34.5^\circ) \approx 0.5664\)

Then \(22 \times 0.901 = 19.822\), divided by 0.5664 is approximately…

Answer:

\(35.00\) (or approximately 35.00, depending on rounding)