Hello, checkmarks!

]1. A helicopter hovers directly above the landing pad on the roof of a 125 m high building.

A person is standing 145 m from the base of the building.

The angle of elevation of the helicopter from this person is 58°.

How high is the helicopter hovering about the landing pad? Code:

*H
* |
* | y
* |
* *A
* |
* |
* | 125
* |
* 58° |
O* - - - - - - - - - *B
145

This one doesn't require the Law of Cosines.

The observer is at $\displaystyle O$. .The helicopter is at $\displaystyle H$.

The building is $\displaystyle AB = 125$. .$\displaystyle OB = 145$.

Let $\displaystyle y = HA$

In right triangle $\displaystyle HBO$, we have: .$\displaystyle \tan58^o \:=\:\frac{y + 125}{145}$

Therefore: .$\displaystyle y \;=\;145\tan58^o - 125 \;\approx\;107$ m.

2. One ship leaves a port and sails at 17 km/h on a bearing of 024 degrees.

A second ship leaves the same port at the same time and sails at 21 km/h

on a bearing of 071 degrees.

How far apart are the two ships after 2 hours?

The bearing is measured clockwise from North.

After two hours, the positions of the ships look like this: Code:

A
: *
: * *
: * * x
: * *
: * *
: *34 *
: * * B
:24° * *
: * *
: * 47° * 42
: * *
:* *
*
P

Now we can use the Law of Cosines.

$\displaystyle x^2\;=\;34^2 + 42^2 - 2(34)(42)\cos47^o \:=\:972.2126837$

Therefore: .$\displaystyle x \;\approx\;31.2$ km.