Question

find angle b given the measurements of the triangle below. round your answers to the nearest degree. identify if the law of sines or law of cosines is used to solve this. angle b = which law is used? choose... law of sines law of cosines identify the exact va... (typo) select one: a. -√2 149° 92° 108° (triangle with sides 12 mi, 14 mi, 21 mi, vertices a, b, c)

Explanation:

Step1: Identify the law

We have a triangle with sides \(a = 14\), \(b = 12\), \(c = 21\) (assuming standard notation, angle \(B\) is opposite side \(AC = 21\), side \(AB = 14\), side \(BC = 12\)). To find angle \(B\) when we know all three sides, we use the Law of Cosines. The formula for Law of Cosines to find angle \(B\) is \(\cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac}\) (wait, correction: Let's define the sides properly. Let \(AB = c = 14\), \(BC=a = 12\), \(AC = b=21\). Then angle \(B\) is between \(AB\) and \(BC\), so the Law of Cosines formula is \(b^{2}=a^{2}+c^{2}-2ac\cos B\). So \(\cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac}\)

Step2: Substitute the values

\(a = 12\), \(c = 14\), \(b = 21\)

\(\cos B=\frac{12^{2}+14^{2}-21^{2}}{2\times12\times14}=\frac{144 + 196- 441}{336}=\frac{340 - 441}{336}=\frac{- 101}{336}\approx - 0.3006\)

Step3: Find angle \(B\)

\(B=\cos^{-1}(-0.3006)\approx108^\circ\) (rounded to nearest degree)

Answer:

Law of Cosines; \(108^\circ\)