Question

Find the longer leg of the triangle.

A. 3

B.
`√(3)`

C. 9

D.
`√(6)`

Answer 1

A

Explanation:

Since the triangle is right use the cosine ratio

cos30° =
`(adjacent )/(hypotenuse)` =
`(adj)/(2√(3) )`, so

`(√(3) )/(2)` =
`(adj)/(2√(3) )`

Multiply both sides by 2
`√(3)`

adj =
`(√(3) )/(2)` × 2
`√(3)` = 3

Answer 2

Choice A. 3.

Explanation:

The triangle in question is a right triangle.

• The length of the hypotenuse (the side opposite to the right angle) is given.
• The measure of one of the acute angle is also given.

As a result, the length of both legs can be found directly using the sine function and the cosine function.

Let
`\text{Opposite}` denotes the length of the side opposite to the
`30^(\circ)` acute angle, and
`\text{Adjacent}` be the length of the side next to this
`30^(\circ)` acute angle.

`\displaystyle \begin{aligned}\text{Opposite} &= \text{Hypotenuse} * \sin{30^(\circ)}\\ &=2√(3)* (1)/(2) \\&= √(3)\end{aligned}`.

Similarly,

`\displaystyle \begin{aligned}\text{Adjacent} &= \text{Hypotenuse} * \cos{30^(\circ)}\\ &=2√(3)* (√(3))/(2) \\&= 3\end{aligned}`.

The longer leg in this case is the one adjacent to the
`30^(\circ)` acute angle. The answer will be
`3`.

There's a shortcut to the answer. Notice that
`\sin{30^(\circ)} < \cos{30^(\circ)}`. The cosine of an acute angle is directly related to the adjacent leg. In other words, the leg adjacent to the
`30^(\circ)` angle will be the longer leg. There will be no need to find the length of the opposite leg.

Does this relationship
`sin(\theta) < cos(\theta)` holds for all acute angles? (That is,
`0^(\circ) < \theta <90^(\circ)`?) It turns out that:

• 
  `sin(\theta) < cos(\theta)` if
  `0^(\circ) < \theta <45^(\circ)`;
• 
  `sin(\theta) > cos(\theta)` if
  `45^(\circ) < \theta <90^(\circ)`;
• 
  `sin(\theta) = cos(\theta)` if
  `\theta = 45^(\circ)`.