The Law of Cosines
Let a,b, and c be the length of the sides of a given triangle, and x the included angle between sides a and b, then the following relation applies:
[tex]c^2=a^2+b^2-2ab\cos x[/tex]
The triangle shown in the figure has two side lengths of a=4 and b=5. The included angle between them is x=100°. We can find the side length c by substituting the given values in the formula:
[tex]c^2=4^2+5^2-2\cdot4\cdot5\cos 100^o[/tex]
Calculating:
[tex]c^2=16+25-40\cdot(-0.17365)[/tex][tex]\begin{gathered} c^2=47.946 \\ c=\sqrt[]{47.946}=6.92 \end{gathered}[/tex]
Now we can apply the law of the sines:
[tex]\frac{4}{\sin A}=\frac{5}{\sin B}=\frac{c}{\sin 100^o}[/tex]
Combining the first and the last part of the expression above:
[tex]\begin{gathered} \frac{4}{\sin A}=\frac{c}{\sin100^o} \\ \text{Solving for sin A:} \\ \sin A=\frac{4\sin100^o}{c} \end{gathered}[/tex]
Substituting the known values:
[tex]\begin{gathered} \sin A=0.57 \\ A=\arcsin 0.57=34.7^o \end{gathered}[/tex]
The last angle can be ob
