Question

the figure shows the terminal ray of angle $\theta$, in standard position, intersecting the unit circle at point $p$ in the $xy$-plane. the function $g$ is given by $g(x) = \cos x$. for the angle $\omega$ (not shown), $\theta < \omega < \pi$. which of the following is true?

a $g(\omega) < g(\theta)$

b $g(\omega) > g(\theta)$

c $g(\omega) = g(\theta)$

d depending on the value of $\omega$, sometimes $g(\omega) = g(\theta)$ and sometimes $g(\omega) > g(\theta)$

Explanation:

Step1: Analyze cosine function trend

The cosine function $g(z)=\cos z$ is decreasing on the interval $(0, \pi)$. This means as the input angle increases within this range, the output value of the cosine function decreases.

Step2: Apply given angle inequality

We are told $\theta < \omega < \pi$. Since $\cos z$ decreases on $(0, \pi)$, a larger input angle gives a smaller output value.

Answer:

A. $g(\omega) < g(\theta)$