Question

solve for b.
a = 8
b = ?
law of sines \frac{\sin a}{a}=\frac{\sin b}{b}=\frac{\sin c}{c}
round your answer to the nearest hundredth

Explanation:

Step1: Find angle C

In a triangle, the sum of angles is \(180^\circ\). So, \(\angle C=180^\circ - 82^\circ - 55^\circ = 43^\circ\). Wait, no, wait. Wait, the Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Wait, in the triangle, side \(a\) is opposite angle \(A\)? Wait, no, in standard notation, side \(a\) is opposite angle \(A\), side \(b\) opposite angle \(B\), side \(c\) opposite angle \(C\). Wait, in the diagram, angle at B is \(82^\circ\), angle at A is \(55^\circ\), so side \(a\) is opposite angle \(A\)? Wait, no, wait, the label: side \(a = 8\) is at B, so side \(a\) is opposite angle \(A\)? Wait, no, let's check the notation. In triangle \(ABC\), side \(a\) is opposite angle \(A\), side \(b\) opposite angle \(B\), side \(c\) opposite angle \(C\). Wait, in the diagram, side \(a = 8\) is between B and C? Wait, no, the diagram: points B, A, with angle at B is \(82^\circ\), angle at A is \(55^\circ\), side \(a = 8\) is from B to C, so side \(a\) is opposite angle \(A\) (angle at A is \(55^\circ\)), side \(b\) is from A to C, opposite angle \(B\) (angle at B is \(82^\circ\)). Wait, so angle at B is \(82^\circ\), angle at A is \(55^\circ\), so angle at C is \(180 - 82 - 55 = 43^\circ\)? Wait, no, wait, maybe I mixed up. Wait, the Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}\). Wait, angle A is \(55^\circ\), angle B is \(82^\circ\), side \(a\) (opposite angle A) is 8? Wait, no, in the diagram, side \(a = 8\) is adjacent to angle B and angle C? Wait, no, the label: \(a = 8\) is at B, so side \(a\) is BC, which is opposite angle A (angle at A is \(55^\circ\)). So side \(a = 8\) (BC) is opposite angle A (\(55^\circ\)), side \(b\) (AC) is opposite angle B (\(82^\circ\)). So using Law of Sines: \(\frac{\sin A}{a}=\frac{\sin B}{b}\). Wait, no, Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\), where \(a\) is opposite angle \(A\), \(b\) opposite angle \(B\), \(c\) opposite angle \(C\). So angle A is \(55^\circ\), opposite side \(a = 8\)? Wait, no, that can't be, because in the diagram, side \(a = 8\) is from B to C, so angle at A is \(55^\circ\), so side BC (a) is opposite angle A. So angle A: \(55^\circ\), opposite side a = 8. Angle B: \(82^\circ\), opposite side b (AC). So we can use \(\frac{\sin A}{a}=\frac{\sin B}{b}\). Wait, no, wait, \(\frac{\sin A}{a}=\frac{\sin B}{b}\) would be \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\)? Wait, no, that would be if a is opposite A, but maybe I got the sides reversed. Wait, let's re-express. Let's denote:

- Angle at A: \(55^\circ\) (angle A)
- Angle at B: \(82^\circ\) (angle B)
- Therefore, angle at C: \(180 - 55 - 82 = 43^\circ\) (angle C)
- Side a: opposite angle A, so side a is BC, length 8
- Side b: opposite angle B, so side b is AC, which we need to find
- Side c: opposite angle C, which is AB

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

Wait, no, Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\), so \(\frac{\sin A}{a}=\frac{\sin B}{b}\)

So \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\)? Wait, no, that would be if a is opposite A, but if side a is 8, opposite angle A (55°), and side b is opposite angle B (82°), then:

\(\frac{\sin A}{a}=\frac{\sin B}{b}\)

So \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\)? Wait, no, that would give \(b=\frac{8 \sin 82^\circ}{\sin 55^\circ}\), but wait, no, wait, maybe I mixed up the sides. Wait, no, wait, the problem says "Solve for b", and the Law of Sines is given as \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Wait, in the diagram, side \(a = 8\) is at B, so side \(a\) is BC, angle at A is \(55^\circ\), so angle A is opposite side BC (a = 8). Angle at B is \(82^\circ\), opposite side AC (b). So yes, so \(\frac{\sin A}{a}=\frac{\sin B}{b}\)

So \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\)

Wait, no, wait, that would be \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\), so solving for b: \(b = \frac{8 \sin 82^\circ}{\sin 55^\circ}\)

Wait, but let's check the angles again. Wait, maybe I made a mistake in the angle opposite. Wait, let's recast:

In triangle ABC:

- Angle at A: \(55^\circ\) (angle A)
- Angle at B: \(82^\circ\) (angle B)
- Therefore, angle at C: \(180 - 55 - 82 = 43^\circ\) (angle C)
- Side a: opposite angle A (BC) = 8
- Side b: opposite angle B (AC) =?
- Side c: opposite angle C (AB)

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

So \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\)

Wait, no, that would be \(\frac{\sin 55^\circ}{8}=\frac{\sin 82^\circ}{b}\), so \(b = \frac{8 \sin 82^\circ}{\sin 55^\circ}\)

Wait, but let's calculate that.

First, calculate \(\sin 82^\circ\) and \(\sin 55^\circ\)

\(\sin 82^\circ \approx 0.9903\)

\(\sin 55^\circ \approx 0.8192\)

So \(b = \frac{8 \times 0.9903}{0.8192} \approx \frac{7.9224}{0.8192} \approx 9.67\)

Wait, but wait, maybe I mixed up the sides. Wait, maybe side \(a\) is opposite angle B? Wait, no, the label says \(a = 8\) is at B, so side \(a\) is BC, angle at A is \(55^\circ\), so BC is opposite angle A. So that should be correct.

Wait, alternatively, maybe the Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}\), where angle A is \(55^\circ\), side \(a\) is 8, angle B is \(82^\circ\), side \(b\) is what we need. So:

\(b = \frac{a \sin B}{\sin A}\)

Yes, that's the correct formula. So \(a = 8\), \(\sin B = \sin 82^\circ\), \(\sin A = \sin 55^\circ\)

So \(b = \frac{8 \times \sin 82^\circ}{\sin 55^\circ}\)

Calculating:

\(\sin 82^\circ \approx 0.9902680687\)

\(\sin 55^\circ \approx 0.8191520443\)

So \(b = \frac{8 \times 0.9902680687}{0.8191520443} \approx \frac{7.92214455}{0.8191520443} \approx 9.67\)

So rounding to the nearest hundredth, \(b \approx 9.67\)

Wait, but let's check again. Wait, maybe the angle at B is \(82^\circ\), angle at A is \(55^\circ\), so angle at C is \(180 - 82 - 55 = 43^\circ\). Then side \(a = 8\) is opposite angle A (55°), side \(b\) opposite angle B (82°), so Law of Sines: \(\frac{a}{\sin A} = \frac{b}{\sin B}\), so \(b = \frac{a \sin B}{\sin A}\)

Yes, that's correct. So:

\(b = \frac{8 \times \sin 82^\circ}{\sin 55^\circ}\)

Calculating:

\(\sin 82^\circ \approx 0.9903\)

\(\sin 55^\circ \approx 0.8192\)

\(8 \times 0.9903 = 7.9224\)

\(7.9224 / 0.8192 \approx 9.67\)

So the value of \(b\) is approximately 9.67.

Step2: Calculate the value of b

Using the Law of Sines formula \(\frac{\sin A}{a}=\frac{\sin B}{b}\), we rearrange to solve for \(b\):

\(b = \frac{a \cdot \sin B}{\sin A}\)

Substitute \(a = 8\), \(\angle A = 55^\circ\), \(\angle B = 82^\circ\):

\(b = \frac{8 \cdot \sin(82^\circ)}{\sin(55^\circ)}\)

Calculate \(\sin(82^\circ) \approx 0.9903\) and \(\sin(55^\circ) \approx 0.8192\):

\(b = \frac{8 \cdot 0.9903}{0.8192} \approx \frac{7.9224}{0.8192} \approx 9.67\)

Answer:

\(b \approx \boxed{9.67}\)