Answer:
x = 6
Step-by-step explanation:
Right Triangle
We are given a right triangle whose shortest side has a length of 3 units. This side must be opposite to the smallest acute angle of 30°.
The triangle is shown in the figure attached.
The tangent ratio relates the opposite side with the adjacent side. The formula can be applied to the angle of 30° as follows:
[tex]\displaystyle \tan 30^\circ=\frac{\text{opposite leg}}{\text{adjacent leg}}[/tex]
[tex]\displaystyle \tan 30^\circ=\frac{3}{y}[/tex]
Solving for y:
[tex]\displaystyle y=\frac{3}{\tan 30^\circ}[/tex]
Since:
[tex]\tan 30^\circ=\frac{1}{\sqrt{3}}[/tex]
[tex]\displaystyle y=3\sqrt{3}[/tex]
Now applying the sine to find the hypotenuse x:
[tex]\displaystyle \sin 30^\circ=\frac{\text{opposite leg}}{\text{hypotenuse}}[/tex]
[tex]\displaystyle \sin 30^\circ=\frac{3}{x}[/tex]
Solving for x:
[tex]\displaystyle x=\frac{3}{\sin 30^\circ}[/tex]
Since:
[tex]\sin 30^\circ=\frac{1}{2}[/tex]
[tex]\displaystyle x=\frac{3}{\frac{1}{2}}[/tex]
x = 6
