Question

let the function (f) be defined as (f(\theta) = cos(\theta)) and function (g) be defined as (g(\theta) = sin(\theta)).

describe (f) and (g) on the open interval (\frac{pi}{2} < \theta < pi).

both f and g are increasing.
both f and g are decreasing.
f is increasing and g is decreasing.
f is decreasing and g is increasing.

Explanation:

Identify the interval and functions

Using the Unit Circle Trigonometry knowledge point
The interval is \(\frac{\pi}{2} < \theta < \pi\), which corresponds to Quadrant II. We analyze the functions \(f(\theta) = \cos(\theta)\) and \(g(\theta) = \sin(\theta)\).

Analyze the cosine function

Using the Trigonometric Ratios knowledge point
In Quadrant II, as \(\theta\) increases from \(\frac{\pi}{2}\) to \(\pi\), the \(x\)-coordinate on the unit circle decreases from \(0\) to \(-1\). Thus, \(f(\theta) = \cos(\theta)\) is decreasing.

Analyze the sine function

Using the Trigonometric Ratios knowledge point
In Quadrant II, as \(\theta\) increases from \(\frac{\pi}{2}\) to \(\pi\), the \(y\)-coordinate on the unit circle decreases from \(1\) to \(0\). Thus, \(g(\theta) = \sin(\theta)\) is decreasing.

Compare with options

Using the Unit Circle Trigonometry knowledge point
Both \(f\) and \(g\) are decreasing on the given interval. This matches the second option.

Answer:

• Both f and g are increasing.
• Both f and g are decreasing. (Correct answer)
• f is increasing and g is decreasing.
• f is decreasing and g is increasing.