Problem 22 - Algebra and Trigonometry

Question

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=\frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the angle

We start by identifying the given angle, which is \(t = \frac{3\pi}{2}\). This angle is measured in radians.

Step 2: Locate the angle on the unit circle

The angle \(\frac{3\pi}{2}\) corresponds to three-quarters of a full rotation around the unit circle. Since a full rotation around the circle is \(2\pi\), dividing this into quarters results in \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\), corresponding to the angles. So, \(\frac{3\pi}{2}\) points directly downward along the negative y-axis.

Step 3: Determine the coordinates on the unit circle

On the unit circle, the terminal point associated with the angle \(\frac{3\pi}{2}\) is at the coordinates \((0, -1)\). This is because the unit circle has a radius of 1 and \(\frac{3\pi}{2}\) is at the bottom of the circle, where x equals 0 and y equals -1.

Step 4: Confirm the terminal point

We confirm that the terminal point \(P(x, y)\) for \(t = \frac{3\pi}{2}\) on the unit circle is indeed \((0, -1)\) because any point on the unit circle has coordinates \((\cos(t), \sin(t))\) and \(\cos\left(\frac{3\pi}{2}\right) = 0\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\).

Key Concepts

Radians

Radians are a unit of measurement for angles, just like degrees. However, they offer a natural way of describing angles, based on the radius of a circle. Imagine taking the radius of a circle and wrapping it around the circle's circumference. The length of that wrap around is what we call 1 radian.

• One complete revolution around a circle is equal to \(2\pi\) radians.
• This is a direct relation with the circle’s circumference, which is \(2\pi r\), but since the radius \(r=1\) for a unit circle, it simplifies to \(2\pi\).

Using radians makes it easier to analyze and perform calculations in trigonometry. That's why you'll often encounter radians when you're working with circles in mathematics.

Terminal Point

The terminal point on the unit circle is essentially the endpoint of an angle's circumference path. When you draw an angle from the circle's center and rotate it counterclockwise, where it lands on the circle is the terminal point. For the angle \(t = \frac{3\pi}{2}\):

• The path goes counterclockwise around the circle.
• It covers three-quarters of the circle's circumference.
• This lands us exactly on the negative y-axis, ending at point \((0, -1)\).

Each terminal point has specific coordinates which are crucial for trigonometric functions and equations.

Coordinates on the Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. Each point on this circle corresponds to an angle and is represented in terms of \(\cos\) and \(\sin\). These functions give you the x and y coordinates, respectively.For any angle \(t\) on the unit circle:

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

For \(t = \frac{3\pi}{2}\):

• \(\cos(\frac{3\pi}{2}) = 0\)
• \(\sin(\frac{3\pi}{2}) = -1\)

So the terminal point \((0, -1)\) is reached, translating to zero in the x-direction and negative one in the y-direction.

Negative Y-axis

The negative y-axis is a specific part of the coordinate system's y-axis. It lies below the origin (0,0). On the unit circle, rotations that point directly downward align with the negative y-axis.For the angle \(t = \frac{3\pi}{2}\):

• It starts from the positive x-axis and rotates counterclockwise.
• After covering three-quarters of the circle, it lands on the negative y-axis.

At this point, the coordinates are \((0, -1)\).This axis is crucial in trigonometry as it marks the boundary for rotation and helps identify key angles and their respective coordinates on the circle.