8.5 - Solving Acute Triangle Problems

Reid’s hot-air balloon is 750.0 m directly above a highway.When Reid is looking west, the angle of depression to Exit85 is 75o. Exit 83 is located 2 km to the east of Exit 85.Determine the angle of depression, to the nearest degree,from the balloon to Exit 83.

75o

2 km

Example #1 cont’d75o

2 km75o

75o

750.0 m

x

75o

x = 2000 m

z = 776.46 m

y

𝜃

𝜃

𝜃

𝑦 2=𝑥2+ 𝑧2−2𝑥𝑧 cos𝑌

Y

X

Z

𝑦 2=(2000)2+(776.46)2−2(2000)(776.46 )cos 75𝑜

𝑦 2=4×106+6.03×105−8.04×105

𝑦 2=3.799×106𝑦=1949

Example #1 cont’d

75o

x = 2000 m

z = 776.46 m

y = 1949 m

𝜃Y

X

Z

We are looking for Angle Z.We can use Sine Law since we have a Side-Angle pair (Y-y).

We can also use both forms of cosine law.

Therefore, the angle of depression from the balloon to Exit 83 is about 23o.

Example #2N

The captain of a boat leaves a marina and heads due west for 25 km. Then the captain adjuststhe course of his boat and heads N 30o E for 20 km. How far is the boat from the marina?

We are looking for ‘r’.

𝑟2=(20)2+(25)2−2 (20)(25)cos60𝑜

Therefore, the boat is about 23 km away from the marina.