Question

the ray y = x, x ≥ 0 contains the origin and all points in the coordinate system whose bearing is 45°. determine the equation of a ray consisting of the origin and all points whose bearing is 30°. the equation of the ray is y = □, x ▼ 0. (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall slope from bearing

A bearing of \(30^\circ\) from the positive x - axis (since the ray starts at the origin and we consider the angle with the x - axis) means the slope \(m\) of the line is \(\tan(30^\circ)\). We know that \(\tan(\theta)=\frac{y}{x}\) for a line passing through the origin \(y = mx\), where \(\theta\) is the angle made with the positive x - axis. And \(\tan(30^\circ)=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\) (rationalizing the denominator).

Step2: Determine the equation of the ray

Since the ray starts at the origin \((0,0)\) and has a slope of \(\tan(30^\circ)=\frac{\sqrt{3}}{3}\), and \(x\geq0\) (because the ray consists of the origin and points with non - negative x - values, similar to the ray \(y = x,x\geq0\)), the equation of the ray is \(y=\frac{\sqrt{3}}{3}x\) with \(x\geq0\).

Answer:

The equation of the ray is \(y = \frac{\sqrt{3}}{3}x\), \(x\geq0\).