Question

use the information to answer the questions.
which formula can you use to find b?
in triangle abc, a = 11 in., m∠b = 70°, and
\\( \frac { a } { \sin ( a ) } = \frac { b } { \sin ( b ) } \\)
c = 9 in.
\\( c \square \sin ( b ) = b \square \sin ( a ) \\)
which information about the triangle is given?
\\( b ^ { 2 } = a ^ { 2 } + c ^ { 2 } - 2 a c \square \cos ( b ) \\)
\\( b ^ { 2 } = a ^ { 2 } + c ^ { 2 } - 2 a c \square \cos ( b ) \\)

Explanation:

Step1: Recall the Law of Cosines formula

The Law of Cosines formula for a triangle with sides \(a\), \(b\), \(c\) and angle \(B\) is \(b^{2}=a^{2}+c^{2}-2ac\cos(B)\).

Step2: Analyze other options

• The formula \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}\) is the Law of Sines. But we don't know angle \(A\) yet, so it's not straightforward to use it here.
• The formula \(c\sin(B) = b\sin(a)\) is not a valid trigonometric formula for triangles.
• The formula \(B^{2}=a^{2}+c^{2}-2ac\cos(b)\) has incorrect notation (angle \(B\) and side \(b\) are mis - used in terms of squaring and cosine argument).

Answer:

\(b^{2}=a^{2}+c^{2}-2ac\cos(B)\)