Question

check both the sin (t) and cos (t) check boxes, and the approximate values check box. use the t - slider to change t from 0 to \\( \frac { \pi } { 2 } \\). as t increases from 0 to \\( \frac { \pi } { 2 } \\), the x - coordinate __ and the y - coordinate __
use the interactive figure to find your answer. use the left and right arrow keys to move along a slider as needed.
click here to launch the interactive figure.
as t increases from 0 to \\( \frac { \pi } { 2 } \\), the x - coordinate
and the y - coordinate

Explanation:

Step1: Recall the unit - circle definitions

On the unit circle, for a point \((x,y)\) corresponding to an angle \(t\), \(x = \cos(t)\) and \(y=\sin(t)\).

Step2: Analyze the behavior of \(\cos(t)\) as \(t\) increases from \(0\) to \(\frac{\pi}{2}\)

The cosine function \(y = \cos(t)\) has the formula \(\cos(t)=\frac{x}{r}\) (on the unit circle \(r = 1\)). As \(t\) increases from \(0\) to \(\frac{\pi}{2}\), using the formula \(\cos(t)\) is a decreasing function. We know that \(\cos(0)=1\) and \(\cos(\frac{\pi}{2}) = 0\).

Step3: Analyze the behavior of \(\sin(t)\) as \(t\) increases from \(0\) to \(\frac{\pi}{2}\)

The sine function \(y=\sin(t)\) has the formula \(\sin(t)=\frac{y}{r}\) (on the unit circle \(r = 1\)). As \(t\) increases from \(0\) to \(\frac{\pi}{2}\), \(\sin(t)\) is an increasing function. We know that \(\sin(0)=0\) and \(\sin(\frac{\pi}{2})=1\).

Answer:

The \(x -\)coordinate (which is \(\cos(t)\)) decreases and the \(y -\)coordinate (which is \(\sin(t)\)) increases.