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Chapter 6: Problem 75

If you are given two sides of a triangle and their included angle, you can find the triangle's area. Can the Law of Sines be used to solve the triangle with this given information? Explain your answer.

Short Answer

Expert verified
No, the Law of Sines cannot directly be applied to solve a triangle when two sides and their included angle are known. Instead, the Rule of Area for a triangle would be applicable in this case.

Step by step solution

01

Understanding what the Law of Sines is

The Law of Sines states that in any triangle, the ratio of the length of any side to the sine of the angle opposite that side is the same for all three sides of the triangle. This can be formulated as \( \frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC} \) where a, b, c are the sides and A, B, C are the opposite angles respectively.
02

Applicability of Law of Sines

In situations where the measure of two sides and the angle between them are given, also known as the Side-Angle-Side (SAS) condition, the Law of Sines may not immediately apply, because for the Law of Sines to be used, either two angles and one side (Angle-Angle-Side or AAS) or two sides and an angle opposite one of them (Side-Side-Angle or SSA) should be known.
03

Finding the area of the triangle

For this SAS condition, another formula called the 'Rule of Area of a triangle' can be more applicable. The area of a triangle when two sides and their included angle are known can be calculated by the formula \( Area = \frac{1}{2}ab \sin(C) \) where \( a \) and \( b \) are the lengths of the two sides and \( C \) is the included angle.

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