Question

question 10 of 10
given any triangle abc labeled as shown, which equation cannot be applied to solve for a side or angle using the law of sines?
a. \\( \frac { \sin a } { c } = \frac { \sin c } { a } \\)
b. \\( \frac { \sin b } { b } = \frac { \sin c } { c } \\)
c. \\( \frac { c } { \sin c } = \frac { a } { \sin a } \\)
d. \\( \frac { a } { \sin a } = \frac { b } { \sin b } \\)

Explanation:

Step1: Recall the Law of Sines formula

The Law of Sines states that in a triangle \(ABC\) with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).

Step2: Analyze each option

• Option A:

From the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\), cross - multiplying gives \(\frac{\sin A}{c}=\frac{\sin C}{a}\), which is incorrect as it should be \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

• Option B:

From \(\frac{b}{\sin B}=\frac{c}{\sin C}\), we can rewrite it as \(\frac{\sin B}{b}=\frac{\sin C}{c}\) (by taking reciprocals). This is a valid form of the Law of Sines.

• Option C:

From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can rewrite it as \(\frac{c}{\sin C}=\frac{a}{\sin A}\). This is a valid form of the Law of Sines.

• Option D:

From \(\frac{a}{\sin A}=\frac{b}{\sin B}\). This is a direct application of the Law of Sines.

Answer:

A. \(\frac{\sin A}{c}=\frac{\sin C}{a}\)