Answer:
Choice A. 3.
Step-by-step explanation:
The triangle in question is a right triangle.
As a result, the length of both legs can be found directly using the sine function and the cosine function.
Let [tex]\text{Opposite}[/tex] denotes the length of the side opposite to the [tex]30^{\circ}[/tex] acute angle, and [tex]\text{Adjacent}[/tex] be the length of the side next to this [tex]30^{\circ}[/tex] acute angle.
[tex]\displaystyle \begin{aligned}\text{Opposite} &= \text{Hypotenuse} \times \sin{30^{\circ}}\\ &=2\sqrt{3}\times \frac{1}{2} \\&= \sqrt{3}\end{aligned}[/tex].
Similarly,
[tex]\displaystyle \begin{aligned}\text{Adjacent} &= \text{Hypotenuse} \times \cos{30^{\circ}}\\ &=2\sqrt{3}\times \frac{\sqrt{3}}{2} \\&= 3\end{aligned}[/tex].
The longer leg in this case is the one adjacent to the [tex]30^{\circ}[/tex] acute angle. The answer will be [tex]3[/tex].
There's a shortcut to the answer. Notice that [tex]\sin{30^{\circ}} < \cos{30^{\circ}}[/tex]. The cosine of an acute angle is directly related to the adjacent leg. In other words, the leg adjacent to the [tex]30^{\circ}[/tex] angle will be the longer leg. There will be no need to find the length of the opposite leg.
Does this relationship [tex]\sin{\theta} < \cos{\theta}[/tex] holds for all acute angles? (That is, [tex]0^{\circ} < \theta <90^{\circ}[/tex]?) It turns out that:
Answer:
A
Step-by-step explanation:
Since the triangle is right use the cosine ratio
cos30° = [tex]\frac{adjacent }{hypotenuse}[/tex] = [tex]\frac{adj}{2\sqrt{3} }[/tex], so
[tex]\frac{\sqrt{3} }{2}[/tex] = [tex]\frac{adj}{2\sqrt{3} }[/tex]
Multiply both sides by 2[tex]\sqrt{3}[/tex]
adj = [tex]\frac{\sqrt{3} }{2}[/tex] × 2[tex]\sqrt{3}[/tex] = 3
