Question

when finding \\( cos \theta = \frac { sqrt { 3 } } { 2 } \\) on a unit circle, what type of special right triangle or a ray is used? (1 point) 30 - 60 - 90 triangle a horizontal ray from the origin 45 - 45 - 90 triangle a vertical ray from the origin

Explanation:

We know that in a unit circle, for a \(30 - 60-90\) triangle, the cosine of \(30^{\circ}\) is \(\frac{\sqrt{3}}{2}\). In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). The cosine of an angle in a right - triangle (in the context of the unit circle) is given by \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta = 30^{\circ}\), if the hypotenuse \(r = 1\) (unit circle), and the adjacent side \(x=\frac{\sqrt{3}}{2}\).

A horizontal ray from the origin would correspond to \(\theta = 0^{\circ}\) or \(\theta=180^{\circ}\) with \(\cos\theta=\pm1\). For a \(45 - 45-90\) triangle, \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\). A vertical ray from the origin would correspond to \(\theta = 90^{\circ}\) or \(\theta = 270^{\circ}\) with \(\cos\theta = 0\).

Answer:

30 - 60 - 90 triangle