Given the right triangle of the figure, we must compute the length of the sides AB and AC.
From the picture we read the following data about the triangle:
- we consider the angle
[tex]\theta=60^{\circ}[/tex]- respect to the angle θ we see that the adjacent cathetus side is BC, and its length is:
[tex]BC=6[/tex]- respect to the angle θ the opposite cathetus is AC
- the hypotenuse of the triangle is AB
We have the following trigonometric identities:
[tex]\begin{gathered} (1)\rightarrow\cos \theta=\frac{\text{adjacent cathetus}}{\text{hypotenuse}} \\ (2)\rightarrow\tan \theta=\frac{\text{opposite cathetus}}{\text{adjacent cathetus}} \end{gathered}[/tex]1) Using equation (1) and the data above we compute the value of the hypotenuse AB:
[tex]\begin{gathered} \cos 60^{\circ}=\frac{6}{AB} \\ AB\cdot\cos 60^{\circ}=6 \\ AB=\frac{6}{\cos 60^{\circ}} \\ AB=\frac{6}{0.5}=12 \end{gathered}[/tex]2) Using equation (2) and the data above we compute the value of the opposite cathetus AC:
[tex]\begin{gathered} \tan 60^{\circ}=\frac{AC}{6} \\ AC=\tan 60^{\circ}\cdot6 \\ AC=\sqrt[]{3}\cdot6 \end{gathered}[/tex]Answers
AB = 12
AC = 6√3