Question

an angle measures θ radians (where θ > 0). a circle is centered at the angles vertex, and the terminal point is highlighted in purple. given that cos(θ) = -0.575 and sin(θ) = -0.818, complete the following. a. what is the terminal points vertical distance above the circles center in units of the circles radius? b. what is the terminal points horizontal distance to the right of the circles center in units of the circles radius?

Explanation:

Step1: Recall the definitions of sine and cosine in the unit - circle

In a unit - circle (where the radius \(r = 1\)), if the terminal point of an angle \(\theta\) is \((x,y)\), then \(\cos\theta=x\) and \(\sin\theta = y\). The vertical distance \(y\) (a) and the horizontal distance \(x\) (b) are related to the trigonometric functions.

Step2: Find the vertical distance (a)

We know that for a point \((x,y)\) on the unit - circle (radius \(r = 1\)), the vertical distance of the terminal point above the center of the circle is given by the sine function. Since \(\sin\theta=- 0.818\), the vertical distance \(a=-0.818\) (negative because it is below the center in the context of the trigonometric sign; if we consider the magnitude in terms of radius lengths, \(|a| = 0.818\)).

Step3: Find the horizontal distance (b)

We know that for a point \((x,y)\) on the unit - circle (radius \(r = 1\)), the horizontal distance of the terminal point to the right of the center of the circle is given by the cosine function. Since \(\cos\theta=-0.575\), the horizontal distance \(b = - 0.575\) (negative because it is to the left of the center in the context of the trigonometric sign; if we consider the magnitude in terms of radius lengths, \(|b|=0.575\)).

Answer:

a. \(-0.818\)
b. \(-0.575\)