Question

to apply the law of sines to find the length of side ( b ) of the triangle, it is enough to know which of the following?

a. ( a,angle b )
b. ( angle a,angle b,angle c )
c. ( c,angle b,angle c )
d. ( a,c )

Explanation:

Step1: Recall the Law of Sines formula

The Law of Sines is given by \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). To find side \(b\), we need a ratio where we know a side - angle pair (either \(a\) and \(\angle A\), \(b\) and \(\angle B\), or \(c\) and \(\angle C\)) and another angle.

Step2: Analyze option C

If we know \(c\), \(\angle B\), and \(\angle C\):
First, find \(\angle A = 180^{\circ}-(\angle B+\angle C)\) (using the angle - sum property of a triangle \(\angle A+\angle B+\angle C = 180^{\circ}\)).
Then, using the Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\), we can solve for \(b\) as \(b=\frac{c\sin B}{\sin C}\).

Answer:

C. \(c,\angle B,\angle C\)