Answer:
m∠A = 45.86°
Step-by-step explanation:
A rough sketch of the triangle is shown in the attached pic.
When 3 sides are given and we want to solve for an angle, we use the Cosine Rule. Which is:
[tex]p^2=a^2 +b^2 -2abCosP[/tex]
Where a, b, p are the lengths of 3 sides (with p being the side opposite of the angle we are solving for) and P is the angel we want to solve for
Thus, we have:
[tex]p^2=a^2 +b^2 -2abCosP\\13^2=14^2 +18^2-2(14)(18)CosA\\169=520-504CosA\\504CosA=351\\CosA=\frac{351}{504}\\CosA=0.6964\\A=Cos^{-1}(0.6964)=45.86[/tex]
In △ABC,a=13, b=14, and c=18. Then angle, m∠A is is 46.654°
[tex]\frac{a}{sinA}=\frac{b}{sinB} =\frac{c}{sinC}[/tex]
[tex]a^{2} =b^{2} +c^{2} -2bcCosA[/tex] or
[tex]b^{2} =a^{2} +c^{2} -2acCosB[/tex] or
[tex]c^{2} =a^{2} +b^{2} -2abCosC[/tex]
In our case;
we are going to use Cosine rule to find m∠A
We are given;
a=13, b=14, and c=18
Therefore;
[tex]a^{2} =b^{2} +c^{2} -2bcCosA[/tex]
Replacing the variables;
[tex]13^{2} =14^{2} +18^{2} -2(14)(18)CosA[/tex]
Making CosA the subject;
[tex]CosA = \frac{(13^{2} -14^{2} -18^{2})}{-2(14)(18)}[/tex]
[tex]Cos A = \frac{-351}{-504}[/tex]
[tex]CosA = 0.6964[/tex]
[tex]A = Cos^{-1} (0.6864)[/tex]
[tex]A = 46.654[/tex]
Therefore; In △ABC,a=13, b=14, and c=18, m∠A is 46.654°
Keywords: Sine rule, Cosine rule
Level; High school
Subject: Mathematics
Topic: Triangles
Sub-topic: Cosine and sine rule
