two lighthouses are situated such that B is 7 km directly east of lighthouse A. A ship at point P observes that A is due north and that the bearing of B is 46°10'. How far is the ship from A and B ?

According to my sketch:

tan46°10' = 7/AP
AP = 7/ tan46°10' = ....

sin46°10' = 7/BP
BP = 7/sin46°10' = ....

how do you sketch this kind of problem?, that's the reason why i'm having trouble with this ..

first thing, of course, is to draw line AB. B is east of A, so it's horizontal.

A is due north of P, so draw a line south from A and put P somewhere on it.

You can estimate angles, at least to some degree :-) so make PB at about 45 degrees.

To solve this problem, we can use trigonometry. Let's consider point P as the location of the ship.

Step 1: Determining the distance between the ship (P) and lighthouse A

Since the ship observes A due north, we can conclude that the angle between the line connecting P and A (PA) and the north direction is 90°.

Using trigonometry, we can use the sine function to find the length of PA:

sin(90°) = PA / hypotenuse

As the line connecting P and A (PA) is the hypotenuse, we can write:

1 = PA / hypotenuse

Since we don't know the exact length of PA, we can assume it as x:

1 = x / hypotenuse

Simplifying the equation, we get:

hypotenuse = x

So, the distance between the ship (P) and lighthouse A is simply x.

Step 2: Determining the distance between the ship (P) and lighthouse B

To find the distance between the ship (P) and lighthouse B, we need to consider the bearing of B. The bearing of B is given as 46°10'.

The bearing of an object refers to the angle between the line connecting the observer (P) and the object (B) and the north direction, measured in a clockwise direction.

In this case, the angle between the line PB and the north direction is 46°10'.

Step 3: Calculating the distances

We need to use trigonometry again to find the distances.

Based on the information provided, we have a right-angled triangle formed by the points P, B, and A.

Let's consider the angle between PB and the northern direction as theta (θ). Since the bearing of B is 46°10', we can write:

θ = 90° - 46°10' = 43°50'

Using trigonometry, we can use the cosine function to relate the angle θ to the distances PB and PA:

cos(θ) = PB / hypotenuse

Substituting the values, we have:

cos(43°50') = PB / x

To find the distance PB, we need to know the value of x (length of PA). However, we can relate the distances to each other using the given information that B is 7 km directly east of A.

Since B is east of A, the line connecting P and B (PB) is the hypotenuse and line connecting P and A (PA) is the adjacent side of the angle θ in the right-angled triangle.

Using cosine again, we have:

cos(θ) = PB / PA

Substituting the known values, we have:

cos(43°50') = PB / 7 km

Now we have the relationship between PA and PB (PA = x). We can solve this equation to find the value of PB:

PB = 7 km * cos(43°50')

The distance PB is the distance between the ship (P) and lighthouse B.

To find the distance PA (x), we need to solve the equation we obtained in Step 1:

1 = x / hypotenuse

Since hypotenuse = x, we have:

1 = x / x

Therefore, the distance PA (x) is equal to 1 km.

So, the ship is 1 km away from lighthouse A and PB = 7 km * cos(43°50') gives the distance between the ship and lighthouse B.