Question

Look at photo for accurate description and remember to round to the nearest TENTH as needed

Answer 1

$\begin{gathered} 25.3\Rightarrow128.9\text{ \degree} \\ 12.3\Rightarrow22.2\text{ \degree } \\ 15.7\Rightarrow28.8\text{ \degree } \end{gathered}$

Explanation

to solve this we need to use the law of cosine

it says,

$c^2=a^2+b^2-2ab\cos (C)$

then

Step 1

let

$\begin{gathered} c=25.3 \\ a=12.3 \\ b=15.7 \\ m\angle C=C \end{gathered}$

replace

$\begin{gathered} c^2=a^2+b^2-2ab\cos (C) \\ 25.3^2=12.3^2+15.7^2-2(12.3)(15.7)\cos \text{ C} \\ 640.09=151.29+246.49-386.22\cos \text{ C} \\ 640.09=397.78-386.22\cos \text{ C} \\ \text{subtract 397.78 in both sides} \\ 640.09-397.78=\text{-}379.78+397.78-386.22\cos \text{ C} \\ 242.22=-386.22\cos \text{ C} \\ \text{divide both sides by -386.22} \\ \frac{242.22}{-386.22}=\frac{-386.22}{-386.22}\cos \text{ C} \\ -0.6271555072=\cos \text{ C} \\ \text{ inverse cos in both sides} \\ \cos ^{-1}(-0.627155507)=\cos ^{-1}(\cos C) \\ 128.858=C \\ \text{rounded} \\ C=128.9\text{ \degree} \end{gathered}$

images

128.9 °

Step 2

angle across the side 12.3

exacty as the previous step , we just need to reorder, so

$\begin{gathered} b^2=a^2+c^2-2ab\cos (B) \\ \text{replace} \\ 12.3^2=25.3^2+15.7^2-2(25.3)(15.7)\cos B \\ 151.29=640.09+246.49-2(25.3)(15.7)\cos \text{ B} \\ 151.29=886.58-2(25.3)(15.7)\cos \text{ B} \\ \text{subtract 886.58 in both sides} \\ 151.29-886.58=-886.58+886.58-2(25.3)(15.7)\cos \text{ B} \\ -735.29=-794.42\cos B \\ \frac{-735.29}{-794.42}=\cos B \\ 0.9255683391=\cos \text{ B} \\ \cos ^{-1}(0.9255683391)=B \\ 22.2\text{=B} \end{gathered}$

Step 3

finally, the angle across the side 15.7

$\begin{gathered} a^2=b^2+c^2-2ab\cos (B) \\ \text{replace} \\ 15.7^2=12.3^2+25.3^2-2(12.3)(25.3)\cos A \\ 246.49=151.29+640.09-622.38\cos \text{ A} \\ 246.49=791.38-622.38\cos \text{ A} \\ \text{subtract 791.38 in both sides} \\ 246.49-791.38=-791.38+791.38-622.38\cos \text{ A} \\ -547.89=-622.38\cos \text{ A} \\ \frac{-547.89}{-622.38}=\cos \text{ A} \\ 0.8754940711=\cos \text{ A} \\ \cos ^{-1}(0.8754940711)=A \\ 28.84=A \\ 28.8=A \end{gathered}$

I hope this helps you