Problem 23 - Algebra and Trigonometry

Question

23-32 \(\approx\) Find the terminal point \(P(X, y)\) on the unit circle determined by the given value of \(t\) $$ t=\frac{\pi}{2} $$

Step-by-Step Solution

Answer

The terminal point is \((0, 1)\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0) on the coordinate plane. Any angle in radians can be represented on this circle, and the terminal point is the point where the line defining the angle intersects the circle.

Step 2: Introduction to the Angle

The given angle is \( t = \frac{\pi}{2} \). This is in radians, where \( \pi \approx 3.14 \). An angle of \( \frac{\pi}{2} \) radians corresponds to 90 degrees.

Step 3: Locating the Angle on the Unit Circle

On the unit circle, an angle of \( \frac{\pi}{2} \) radians means you move counter-clockwise from the positive x-axis, at 90 degrees. This places the terminal point directly at the top of the unit circle.

Step 4: Determining the Terminal Point Coordinates

At \( \frac{\pi}{2} \), the terminal point on the unit circle is at the coordinates \((0, 1)\). This is because \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \).

Step 5: Conclusion

Therefore, the terminal point \( P(X, y) \) on the unit circle for the angle \( t = \frac{\pi}{2} \) is \( (0, 1) \).

Key Concepts

Terminal Point

The terminal point on the unit circle is the spot where a line drawn from the origin (the center of the circle) at a specific angle intersects the circumference. This terminal point has coordinates

• \( \left( x, y \right) \)

depending on the size of the angle in radians. For any angle on the unit circle, these coordinates are the values of

• \( \cos(t) \)
• \( \sin(t) \)

The unit circle makes it easy to find these coordinates because its radius is always 1. This means that the coordinates of the terminal points directly correspond to the cosine and sine of the angle. In our exercise, with an angle of \( \frac{\pi}{2} \) radians, the terminal point is located at \( (0, 1) \). This is because, at this position, the line drawn intersects the circle precisely at this point.

Radian Measure

Radian measure is a way of expressing angles, often preferred in mathematics and fields related to engineering and physics. Unlike degrees, a circle measured in radians wraps up naturally because

• 1 full circle = \( 2\pi \) radians
• \( \pi \) radians = 180 degrees

This makes

• \( \frac{\pi}{2} \)

a very special point since it equals 90 degrees. The benefit of using radians is that they relate directly to the radius of the circle, providing a neat and elegant relationship between the arc length and the angle. In our exercise, \( t = \frac{\pi}{2} \) is an angle represented in radians, placing it directly on the y-axis at the top of the unit circle.

Trigonometric Functions

Trigonometric functions such as sine and cosine are an essential part of understanding the unit circle and angles. On the unit circle, these functions are defined as the coordinates of terminal points. For any given angle

• \( t \)

the following applies:

• The x-coordinate = \( \cos(t) \)
• The y-coordinate = \( \sin(t) \)

These functions help to describe wave patterns, oscillations, and other phenomena in engineering and science. In our scenario,

• \( \cos(\frac{\pi}{2}) = 0 \)
• \( \sin(\frac{\pi}{2}) = 1 \)

This is why at \( t = \frac{\pi}{2} \), the terminal point has coordinates \( (0, 1) \), perfectly demonstrating the value of these functions at this angle.