Question

If ∠P is given and the values of r and q are given, then explain whether the Law of Sines or the Law of Cosines should be used to solve for p.

Law of Cosines, all sides are known
Law of Cosines, two sides, and the included angle are known
Law of Sines, all sides are known
Law of Sines, two angles and the included side are known

Answer 1

Law of Cosines, two sides, and the included angle are known

Explanation:

Law of Cosines

`c^2=a^2+b^2-2ab \cos C`

where:

• a, b and c are the sides
• C is the angle opposite side c

This rule can be used to find:

1. an unknown side if 2 sides and the included angle is known.
2. an unknown angle if all 3 sides are known.

Law of Sines

`\sf (\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)`

where:

• A, B and C are the angles
• a, b and c are the sides opposite the angles (a is opposite A, etc.)

This rule can be used to find:

• an unknown side when the opposite angle is known and another side and its opposite angle is known.
• an unknown angle when the opposite side is known and another angle and its opposite side is known.

Therefore, as sides r and q are given and the included angle ∠P is known, we can use the Law of Cosines to find side p:

`\implies p^2=r^2+q^2-2rq \cos P`

`\implies p=√(r^2+q^2-2rq \cos P)`

Answer 2

• B) Law of Cosines, two sides, and the included angle are known

Explanation:

Sides r and q form the angle P.

We need to find the value of side p.

According the the information, we have 2 sides and the included angle known and need to find the missing side.

This is possible using the law of cosines:

• 
  `p=√(r^2+q^2-2rq*cos < P)`

Correct choice is B