We're supposed to figure this out using the sine rule , but for some reason i'm lost. can anyone ? ?

The answer is 26.55.

First identify any supplementary and right angles. Between the boat and 40degrees is a supplementary angle (180degrees), so you minus 180 from 40degrees to get the angle where the boat is at and that angle is 140degrees.

Next we will see that the triangle with the lighthouse and 40degrees is a right angle so that means it should add up to 90degrees, so 90-40 = 50Degrees. The angle in front of the light house at the top, but still inside the right triangle is 50degrees.

Next triangle, we are missing the angle next to 50degrees at the top. We know all interior triangles add up to 180, so 180-124-40 = 16degrees.

Now we must find the hypotenuse in order to solve for the height.
For this we will focus only on the obtuse triangle (where boat 24degrees is).

Set it up as sin(16)/28= sin(140)/c
Cross multiply and then it changes to Csin16 = 28sin(140).
Divide by sine16 on both sides. Sine16 cancel out on the left side.
Now you have, c = 28sin140/sin16.
Put that in on a calculator and C will come out as 65.2961.

Last step, we can now find (H)eight.
Formula: C(SInA) = H/C
65.2961*sin(24degrees) = H/65.2961
Multiply both sides by 65.2961 and the right side will cancel out.

Now, H = 65.2961(sin(24degrees)
H = 26.55

Side note:
It will help to label the triangles angles ABC as well as hypotenuse, adjacent, opposite.

Formula used above to find hypotenuse:
SinB/Distance = SinAdjacent angle (opposite)/ C (hypotenuse)

Good Luck, I know it can be confusing, but you can do it.

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