Question

4
select the correct answer.
for a point on the unit circle, if $\theta$ lies in quadrant iv, what could be the value of $cos(\theta)$?
a. $-\frac{3}{5}$
b. $-\frac{sqrt{41}}{5}$
c. $\frac{3}{5}$
d. $\frac{sqrt{41}}{5}$

Explanation:

Step1: Recall cosine - value in quadrants

In the unit - circle, the cosine of an angle $\theta$ is the $x$ - coordinate of the point on the unit circle corresponding to $\theta$. In quadrant IV, the $x$ - coordinate is positive and the $y$ - coordinate is negative.

Step2: Analyze the options

Options A and B are negative values, so they cannot be the cosine of an angle in quadrant IV. Also, since the point is on the unit - circle, the value of $\cos(\theta)$ and $\sin(\theta)$ satisfy $\cos^{2}(\theta)+\sin^{2}(\theta) = 1$, and $|\cos(\theta)|\leq1$ and $|\sin(\theta)|\leq1$. The value $\frac{\sqrt{41}}{5}\approx\frac{6.4}{5}=1.28>1$, so option D is not valid. Option C, $\frac{3}{5}$, is a positive value less than 1 and can be the $x$ - coordinate (cosine value) of a point in quadrant IV.

Answer:

C. $\frac{3}{5}$