Problem 23 - Precalculus Mathematics for Calculus

Question

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) of the coordinate plane. Any angle corresponds to a point on this circle.

Step 2: Position of Angle on the Unit Circle

The angle given is \(t = \frac{\pi}{2}\). This angle is measured in radians. \(\frac{\pi}{2}\) radians is equivalent to 90 degrees, which is located at the "top" of the unit circle.

Step 3: Identifying the Coordinates

On the unit circle, the coordinates corresponding to \(\frac{\pi}{2}\) (or 90 degrees) are on the positive y-axis. Thus, the point is \((0, 1)\) because it is directly above the origin at a distance of 1.

Key Concepts

Radians

Radians are a way of measuring angles based on the radius of a circle. Let's dive deeper to understand how they work. Unlike degrees, which divide a circle into 360 parts, radians are based on the circumference of the circle itself. This makes radians particularly useful for calculations involving trigonometric functions because they relate directly to the geometry of the circle.
Here’s how radian measure works:

• One complete revolution around a circle equals the length of its circumference.
• This, by definition, is exactly 2π radians.
• Thus, a half-circle is π radians, a right angle is \(rac{\pi}{2}\) radians, and so on.

To convert between degrees and radians, we utilize the formula: \(\text{radians} = \frac{\pi}{180} \times \text{degrees}\).
Radian measure provides a natural way of thinking about the angles in terms of the circle itself.

Terminal Point

The terminal point is a fundamental concept when dealing with angles and circles. On the unit circle, it becomes easy to visualize and understand.
The terminal point refers to the point where an angle’s terminal arm intersects with the circle. Here’s how you find it:

• Start from the positive x-axis and rotate the point counterclockwise by the angle's measure.
• The location where you stop is the terminal point.

This concept is important in trigonometry as it helps visualize the sine and cosine values.
For the exercise given with \(t = \frac{\pi}{2}\), after rotating the arm from the positive x-axis by this angle, we locate the terminal point at \((0, 1)\), right at the top of the unit circle.

Coordinates on a Circle

Understanding the coordinates on a unit circle is key to mastering trigonometry concepts. As the circle is centered at the origin, every point on its circumference can be described using coordinates that relate directly to trigonometric functions.
Key points about these coordinates are:

• On the unit circle, every point \(P(x, y)\) corresponds to \( ( \cos(t), \sin(t)) \).
• The unit circle has a radius of 1, which means the equation \((x^2 + y^2 = 1)\) always holds true.
• At \(t = \frac{\pi}{2}\), the coordinates are \(x = 0\, \text{and}\, y = 1\), making \(P(0, 1)\).

So, when finding coordinates on a unit circle, you are effectively looking at the cosine and sine values for the angle formed with the x-axis. This clear link to trigonometric functions streamlines complex problems within circular motion and oscillations.