Question

find the length of side z in the picture, below.
(there is a right triangle with one angle 30 degrees, hypotenuse length 1, and the side adjacent to the 30-degree angle is z. )
z =
enter your answer in exact form.

Explanation:

Step1: Identify the trigonometric ratio

We have a right - triangle with an angle of \(30^{\circ}\), the hypotenuse is \(1\), and the side adjacent to the \(30^{\circ}\) angle is \(z\). The cosine of an angle in a right - triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), adjacent side \(= z\), and hypotenuse \(= 1\). So, \(\cos(30^{\circ})=\frac{z}{1}\).

Step2: Recall the value of \(\cos(30^{\circ})\)

We know that \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\). Since \(\cos(30^{\circ}) = z\) (because \(\frac{z}{1}=z\)), we can conclude that \(z=\frac{\sqrt{3}}{2}\).

Answer:

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