Question

multiple choice 1 point
an angle has an initial side of fg and a terminal side of fh, where fg and fh are radii of the unit circle centered at the origin of the xy - coordinate plane represented by point f. if the y - coordinate of point h is $\frac{ - \sqrt{3}}{2}$, what is the measure of $\theta$ to the nearest thousandth of a radian?
(image of a coordinate plane with a unit circle centered at f, initial side fg along the positive x - axis, terminal side fh in the third quadrant, and point h on the terminal side)
options:
1.571
4.712
2.094
3.927
clear my selection

Explanation:

Step1: Recall Unit Circle Sine

On the unit circle, the \( y \)-coordinate of a point \( (x,y) \) on the terminal side of angle \( \theta \) is \( \sin\theta \). So, \( \sin\theta = -\frac{\sqrt{3}}{2} \).

Step2: Determine Quadrant

From the graph, point \( H \) is in the third quadrant (below \( x \)-axis, left of \( y \)-axis). In the unit circle, \( \sin\theta = -\frac{\sqrt{3}}{2} \) occurs at \( \theta = \pi + \frac{\pi}{3}=\frac{4\pi}{3}\approx4.1888 \)? Wait, no—wait, the initial side is \( FG \) (positive \( x \)-axis), terminal side \( FH \). Wait, maybe I misread the quadrant. Wait the graph: \( F \) is origin, \( FG \) is positive \( x \)-axis, \( H \) is in third quadrant? Wait no, the graph shows \( H \) is in the third quadrant? Wait no, the \( y \)-coordinate is negative, so sine is negative. The reference angle for \( \sin\alpha=\frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \). So in third quadrant, \( \theta = \pi + \frac{\pi}{3}=\frac{4\pi}{3}\approx4.1888 \), but that's not an option. Wait, maybe fourth quadrant? Wait no, the options are 1.571 (≈π/2), 4.712 (≈3π/2), 2.094 (≈2π/3), 3.927 (≈5π/4? No, 5π/4≈3.927? Wait 5π/4 is 3.92699, and \( \sin(5\pi/4)=-\frac{\sqrt{2}}{2}\approx -0.707 \), not \( -\frac{\sqrt{3}}{2} \). Wait, maybe I made a mistake. Wait the \( y \)-coordinate is \( -\frac{\sqrt{3}}{2} \), so \( \sin\theta = -\frac{\sqrt{3}}{2} \). The solutions for \( \sin\theta = -\frac{\sqrt{3}}{2} \) are \( \theta = \frac{4\pi}{3} \) (third quadrant) and \( \theta = \frac{5\pi}{3} \) (fourth quadrant). Wait the graph: \( H \) is in the third quadrant? Wait the diagram: \( F \) is origin, \( FG \) is positive \( x \)-axis, \( H \) is in the third quadrant (below \( x \)-axis, left of \( y \)-axis). Wait \( \frac{4\pi}{3}\approx4.188 \), but the options include 4.712 (which is \( \frac{3\pi}{2}\approx4.712 \)), 3.927 (≈5π/4? No, 5π/4≈3.927, but \( \sin(5π/4)=-\frac{\sqrt{2}}{2} \). Wait, maybe the angle is measured from the positive \( x \)-axis, going counterclockwise? Wait no, the terminal side is \( FH \), which is in the third quadrant? Wait, maybe I misread the \( y \)-coordinate. Wait the problem says "the \( y \)-coordinate of point \( H \) is \( -\frac{\sqrt{3}}{2} \)". Wait, maybe the angle is in the third quadrant, but the options: 4.712 is \( \frac{3\pi}{2}\approx4.712 \), but \( \sin(\frac{3\pi}{2})=-1 \), not \( -\frac{\sqrt{3}}{2} \). Wait, maybe the initial side is \( FG \) (positive \( x \)-axis), and the terminal side is in the third quadrant, but maybe I made a mistake. Wait, let's check the options. Wait 3.927 is approximately \( \frac{5\pi}{4} \)? No, \( \frac{5\pi}{4}=3.92699 \), \( \sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}\approx -0.707 \). Wait, 4.712 is \( \frac{3\pi}{2}=4.712 \), \( \sin(\frac{3\pi}{2})=-1 \). Wait, 2.094 is \( \frac{2\pi}{3}\approx2.094 \), \( \sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2} \). 1.571 is \( \frac{\pi}{2}\approx1.571 \), \( \sin(\frac{\pi}{2})=1 \). Wait, maybe the angle is measured clockwise? If we measure clockwise, \( \theta = - \frac{\pi}{3} \), but that's negative. Wait, no, the unit circle: \( \sin\theta = y \)-coordinate. So \( \sin\theta = -\frac{\sqrt{3}}{2} \), so \( \theta = \frac{4\pi}{3} \) (third quadrant) or \( \frac{5\pi}{3} \) (fourth quadrant). \( \frac{4\pi}{3}\approx4.188 \), not an option. \( \frac{5\pi}{3}\approx5.235 \), not an option. Wait, maybe the diagram is different. Wait the options: 3.927 is \( \frac{5\pi}{4}\approx3.927 \), but \( \sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2} \). Wait, maybe the problem has a typo, or I misread the \( y \)-coordinate. Wait, maybe it's \( -\frac{\sqrt{2}}{2} \)? Then \( \sin\theta = -\frac{\sqrt{2}}{2} \), so \( \theta = \frac{5\pi}{4}\approx3.927 \), which is an option. Oh! Maybe the \( y \)-coordinate is \( -\frac{\sqrt{2}}{2} \) instead of \( -\frac{\sqrt{3}}{2} \). That must be a typo, or I misread. Let's check the options: 3.927 is \( \frac{5\pi}{4}\approx3.927 \), and \( \sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2} \). So assuming that, then \( \theta = \frac{5\pi}{4}\approx3.927 \), which is option D (3.927).

Step3: Confirm with Options

Given the options, 3.927 is the closest to \( \frac{5\pi}{4} \) (where \( \sin\theta = -\frac{\sqrt{2}}{2} \)), which might be a misprint for \( -\frac{\sqrt{2}}{2} \) instead of \( -\frac{\sqrt{3}}{2} \). So the measure of \( \theta \) is approximately 3.927.

Answer:

3.927