Problem 21 - Algebra and Trigonometry

Question

\(21-30=\) 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 \( P(0, 1) \) on the unit circle.

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. Any angle \( t \) in radians will correspond to a specific point \( P(x, y) \) on the circumference of the unit circle.

Step 2: Identify the Angle on the Unit Circle

The angle \( t = \frac{\pi}{2} \) radians corresponds to a 90-degree angle. On the unit circle, this means moving \( \frac{\pi}{2} \) radians counterclockwise from the positive x-axis.

Step 3: Determine the Coordinates

At the angle \( \frac{\pi}{2} \), the terminal point on the unit circle is vertically upwards from the origin. This point corresponds to the highest point on the unit circle.

Step 4: Find the Coordinates of Point P

For \( t = \frac{\pi}{2} \), the terminal point \( P(x, y) \) on the unit circle is at the coordinates \( (0, 1) \). This is because at \( \frac{\pi}{2} \) radians, the cosine of the angle is 0 and the sine is 1, hence \( x = \cos(\frac{\pi}{2}) = 0 \) and \( y = \sin(\frac{\pi}{2}) = 1 \).

Key Concepts

Trigonometric Functions

The trigonometric functions, sine and cosine, are fundamental to understanding the position of points on the unit circle. These functions relate an angle in the circle to the horizontal and vertical coordinates of a point on its boundary. In the context of the unit circle, which has a radius of 1, these functions are very straightforward and powerful.

• Sine (\( ext{sin}\)): Corresponds to the vertical coordinate of a point on the unit circle. It tells us how far up or down the point is from the origin.
• Cosine (\( ext{cos}\)): Represents the horizontal coordinate. It indicates how far left or right the point is from the origin.

Example: For an angle of \( t = \frac{\pi}{2} \), \( ext{sin}(t) = 1\) and \( ext{cos}(t) = 0\), so the coordinates are \((0, 1)\). This means at \( \frac{\pi}{2} \), you are directly upwards from the circle's center.

Coordinates of a Point

In the unit circle, every angle \( t \) corresponds to a point \( P(x, y) \) on its boundary. Understanding coordinates is essential because they tell you the exact location of this point in two dimensions. Each coordinate identifies a direction: the x-coordinate for horizontal and the y-coordinate for vertical.

• The x-coordinate is found using \( ext{cos}(t)\). It reveals how far the point is from the vertical axis at the circle's center.
• The y-coordinate is determined by \( ext{sin}(t)\). It shows the point's distance from the horizontal axis passing through the center.

For example, with \( t = \frac{\pi}{2} \), we calculate \( x = 0 \) and \( y = 1 \), placing the point at the top of the circle.

Radian Measure

Radian measure is a way of expressing angles based on the radius of the circle. Unlike degrees, which divide a circle into 360 parts, radians divide it in relation to the circle's circumference. The radian measure is particularly useful in trigonometry because it simplifies many formulas and calculations.

• Basics: One complete revolution around the circle equals \( 2\pi \) radians, which is equivalent to 360 degrees.
• Fractional Relationships: A right angle is \( \frac{\pi}{2} \) radians, which closely relates to our exercise example. This corresponds to moving 90 degrees counterclockwise from the positive x-axis.

Using \( t = \frac{\pi}{2} \), we determine that the terminal point is indeed \((0, 1)\) – directly upwards at a right angle, demonstrating how radians simplify finding points on the unit circle.