To answer this question, we need to understand the properties of the cosine function and the unit circle.
The unit circle is a circle with a radius of 1 centered at the origin of the Cartesian coordinate system. Angles on the unit circle are measured from the positive x-axis, and positive angles are measured counterclockwise.
The cosine of an angle [tex]\(\theta\)[/tex] in the unit circle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.
Quadrant IV is the region of the coordinate plane where the x-values are positive and the y-values are negative. This means that any angle [tex]\(\theta\)[/tex] in quadrant IV will have a positive cosine value because it lies to the right of the y-axis.
Given the options:
A. [tex]\(\frac{3}{5}\)[/tex]
B. [tex]\(-\frac{3}{5}\)[/tex]
C. [tex]\(-\frac{\sqrt{41}}{5}\)[/tex]
D. [tex]\(\frac{\sqrt{41}}{5}\)[/tex]
Since the cosine value is positive in quadrant IV, we can immediately eliminate options B and C, as they are negative.
This leaves us with options A [tex]\(\frac{3}{5}\)[/tex] and D [tex]\(\frac{\sqrt{41}}{5}\)[/tex].
To determine which of these two values is correct, we note that [tex]\(\frac{3}{5}\)[/tex] is a simpler, more common value that often appears as the cosine of certain angles in right triangles that are part of well-known trigonometric ratios often associated with integer lengths of sides.
Therefore, the correct value of [tex]\(\cos(\theta)\)[/tex] in quadrant IV is:
A. [tex]\(\frac{3}{5}\)[/tex]
Thus, the value of [tex]\(\cos(\theta)\)[/tex] when [tex]\(\theta\)[/tex] is in quadrant IV is [tex]\(\frac{3}{5}\)[/tex], which equals 0.6.
