How to Apply Trigonometry to General Triangles?

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

\(cos β = \frac {19^2+11^2-23^2}{2(19)(11)}=\frac{361+121-529}{418}=\frac{-47}{418}=-0.112\)

\( β ≅ 96.45^\circ\)

Then, use the law of sines to find the size of the smallest angle \((α)\):

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

\(\frac {sin\ α }{11}=\frac {sin\ 96.45}{23}\)

\( sin\ α =\frac{11× sin\ 96.45}{23}=\frac {11 × 0.99}{23}=\frac {10.89}{23}=0.47\)

\( α ≅ 28.37^\circ\)

Now, the difference between the largest and smallest angle is:

\( 96.45^\circ\ – 28.37^\circ=68.08 ^\circ\ \)

Solving General Triangles – Example 2:

(SAS) Find the length of the third side of the triangle.

To find side \(c\) use the law of cosines: \(c^2=a^2+b^2-2ab.\cos C\)

\(c^2=31^2+31^2-2(31)(31)(cos 17)=961+961-(1922 × cos 17)=1,922-(1,922 × 0.96)=1,922-1845.12=76.88\)

\(c^2=76.88→ c=\sqrt{76.88}= 8.77\)

Exercises for Solving General Triangles

• (AAS) Find the value of \(x\) in the triangle.

• (ASA) Find the value of \(d\) in the triangle.

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