Question

Could you provide a step by step resolution for this question?

Answer 1

Given:

Angle A = 120 degrees

Side opposite angle C = 150 meters

Side opposite angle B = 275 meters

Find:

Angle B

Solution:

Since we have two sides given and an included angle, we can use cosine law.

Let's look for the length of the side opposite Angle A first.

`a^2=b^2+c^2-2bc\cos A`

where a = length of the side opposite Angle A or side BC

b = side opposite Angle B or Side AC

c = side opposite Angle C or Side AB

A = Angle A

Since we already have the data above, let's plug it in to the formula we have.

`a^2=275^2+150^2-2(275)(150)\cos 120`

Then, solve a.

`\begin{gathered} a^2=75,625+22,500-82,500(-0.5) \\ a^2=98,125+41,250 \\ a^2=139,375 \\ \sqrt[]{a^2}=\sqrt[]{139,375} \\ a\approx373.3296\approx373.33 \end{gathered}`

Hence, the length of side opposite a or Side BC is approximately 373.33 meters.

Now, to solve for Angle B, we can use the sine law.

`(\sin A)/(a)=(\sin B)/(b)`

Let's plug in the value of Angle A, side BC or a, and side AC or b to the formula.

`(\sin120)/(373.3296)=(\sin B)/(275)`

Then, solve for Angle B.

`\begin{gathered} \sin B=(275\sin 120)/(373.3296) \\ \sin B=0.6379268552 \\ B=\sin ^(-1)0.6379268552 \\ B\approx39.64 \end{gathered}`

Therefore, the bearing of ship C from ship B is approximately 40 degrees. (rounded off to the nearest degree)