Question

use the triangle shown to the right to evaluate the following expression. if necessary, express the value without a square root in the denominator by rationalizing the denominator.
cos 30°
cos 30° = □
(type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of cosine

In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For an angle \(\theta\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Identify the adjacent side and hypotenuse for \(30^{\circ}\)

In the given right triangle, for the \(30^{\circ}\) angle, the adjacent side (the side next to the \(30^{\circ}\) angle) has length \(\sqrt{3}\), and the hypotenuse (the side opposite the right angle) has length \(2\).

Step3: Calculate \(\cos30^{\circ}\)

Using the definition of cosine, \(\cos30^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{\sqrt{3}}{2}\). Since the denominator does not have a square root, we don't need to rationalize it further.

Answer:

\(\frac{\sqrt{3}}{2}\)