To determine the value of the cosine of an angle measuring [tex]\(\frac{\pi}{2}\)[/tex] radians, let's consider the position of this angle in the unit circle.
1. Understanding the Unit Circle:
- The unit circle is a circle with a radius of 1 centered at the origin [tex]\((0,0)\)[/tex] on the Cartesian plane.
- Any angle in standard position is measured from the positive [tex]\(x\)[/tex]-axis, rotating counterclockwise.
2. Position of [tex]\(\frac{\pi}{2}\)[/tex] Radians:
- An angle of [tex]\(\frac{\pi}{2}\)[/tex] radians corresponds to a 90-degree angle.
- When you measure 90 degrees (or [tex]\(\frac{\pi}{2}\)[/tex] radians) counterclockwise from the positive [tex]\(x\)[/tex]-axis, you land on the positive [tex]\(y\)[/tex]-axis.
3. Coordinates of the Point on the Unit Circle:
- The coordinates of the point where [tex]\(\frac{\pi}{2}\)[/tex] radians intersects the unit circle are [tex]\((0, 1)\)[/tex].
4. Cosine Function and Coordinates:
- The cosine of an angle in the unit circle is defined as the [tex]\(x\)[/tex]-coordinate of the point on the unit circle corresponding to that angle.
5. Evaluating Cosine at the Angle:
- For the angle [tex]\(\frac{\pi}{2}\)[/tex] radians, the point on the unit circle is [tex]\((0, 1)\)[/tex].
- Therefore, the [tex]\(x\)[/tex]-coordinate at this point is 0.
6. Conclusion:
- The value of the cosine of the angle [tex]\(\frac{\pi}{2}\)[/tex] radians is [tex]\(0\)[/tex].
Thus, the value of the cosine of the angle [tex]\(\frac{\pi}{2}\)[/tex] radians is [tex]\(0\)[/tex].
