Question

Group A is on the north side of the fire and is 800 ft away from group B and 900 feet away from group C. Group A knows that the angle between its two adjacent sides is 89.6. “

The emergency responders want to properly inform the nearby towns of the danger proximity. They will need all three angles. To do this they need some additional information. The emergency responders need to know the angle of Group B and Group C. They also need to know the distance between Groups B and C.

Answer 1

1. Length BC is 1200 ft

2. Angle B is 48.6°

3. Angle C is 41.8°

Explanation:

1. Determination of the length BC

Side opposite C = c = 800 ft

Side opposite B = b = 900 ft

Angle A = 89.6°

Side opposite BC = a =?

Using cosine rule, we can obtain length BC as follow:

a² = b² + c² – 2bc CosA

a² = 900² + 800² – 2 × 900 × 800 × Cos89.6

a² = 810000 + 640000 – 1440000 × 0.007

a² = 1450000 – 10080

a² = 1439920

Take the square root of both side

a = √1439920

a ≈ 1200 ft

Thus, length BC is 1200 ft

2. Determination of angle B

Side opposite B = b = 900 ft

Side opposite A = a = 1200 ft

Angle A = 89.6°

Angle B =?

Angle B can be obtained by using sine rule as follow:

a / Sine A = b / Sine B

1200 / Sine 89.6° = 900 / Sine B

Cross multiply

1200 × Sine B = 900 × Sine 89.6°

1200 × Sine B = 900

Divide both side by 1200

Sine B = 900/1200

Sine B = 0.75

Take the inverse of Sine

B = Sine¯¹ 0.75

B = 48.6°

Thus, angle B is 48.6°

3. Determination of angle C

Side opposite C = c = 800 ft

Side opposite A = a = 1200 ft

Angle A = 89.6°

Angle C =?

Angle C can be obtained by using sine rule as follow:

a / Sine A = c / Sine C

1200 / Sine 89.6° = 800 / Sine B

Cross multiply

1200 × Sine C = 800 × Sine 89.6°

1200 × Sine C = 800

Divide both side by 1200

Sine C = 800/1200

Sine C = 0.6667

Take the inverse of Sine

C = Sine¯¹ 0.6667

C = 48.6°

Thus, angle C is 41.8°