Let
x-------> the side opposite the [tex]30\°[/tex] angle
y------> the side opposite the [tex]60\°[/tex] angle
we know that
If two angles of a triangle measure [tex]30\°[/tex] and [tex]60\°[/tex]
then
the third angle measure [tex]90\°[/tex]
Is a right triangle
Remember that
[tex]sin(30\°)=cos (60\°)=\frac{1}{2} \\\\sin(60\°)=cos(30\°)= \frac{\sqrt{3}}{2} \\ \\tan(30\°)= \frac{\sqrt{3}}{3}\\\\tan(60\°)=\sqrt{3}[/tex]
Statements
case A) The side opposite the [tex]30\°[/tex] angle is longer than the side opposite the [tex]60\°[/tex] angle
The statement is false
Because, the ratio of the side opposite the [tex]30\°[/tex] angle to the side opposite the [tex]60\°[/tex] angle is equal to
[tex]\frac{x}{y} =\frac{1}{\sqrt{3}}[/tex]
so
The side opposite the [tex]30\°[/tex] angle is smaller than the side opposite the [tex]60\°[/tex] angle
case B) The side opposite the [tex]60\°[/tex] angle is longer than the side opposite the [tex]30\°[/tex] angle
The statement is true
Because, the ratio of the side opposite the [tex]60\°[/tex] angle to the the side opposite the [tex]30\°[/tex] angle is equal to
[tex]\frac{y}{x} =\sqrt{3}[/tex]
case C) The sides opposite the [tex]60\°[/tex] angle is twice as long as the side opposite the [tex]30\°[/tex]
The statement is false
Because, the side opposite the [tex]60\°[/tex] angle is equal to
[tex]y=\sqrt{3}x[/tex]
case D) There is no way to compare the sides opposite the angles
The statement is false
Because, we can use the trigonometric functions to be able to compare the ratio of the sides opposite the angles
