Problem 2 - Algebra and Trigonometry

Question

(a) If we mark off a distance \(t\) along the unit circle, starting at \((1,0)\) and moving in a counterclockwise direction, we arrive at the (b) The terminal points determined by \(\pi / 2, \pi,-\pi / 2,2 \pi\) are respectively.

Step-by-Step Solution

Answer

(0,1), (-1,0), (0,-1), and (1,0) respectively.

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. Starting at the point (1, 0) and moving counterclockwise, we measure the angle in radians.

Step 2: Terminal Point for \(t = \pi/2\)

An angle of \(\pi/2\) corresponds to 90 degrees on the unit circle. Moving counterclockwise from (1,0), we reach the top of the circle at point (0,1).

Step 3: Terminal Point for \(t = \pi\)

An angle of \(\pi\) radians corresponds to 180 degrees. Moving counterclockwise from (1,0), this takes us to the leftmost point of the circle: (-1,0).

Step 4: Terminal Point for \(t = -\pi/2\)

An angle of \(-\pi/2\) is a clockwise movement of 90 degrees from (1,0). This takes us to the bottom of the circle at (0,-1).

Step 5: Terminal Point for \(t = 2\pi\)

An angle of \(2\pi\) radians completes a full circle (360 degrees), bringing us back to the original starting point (1,0).

Key Concepts

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. Unlike degrees, where a full circle is divided into 360 parts, in radians, the angle is based on the length of the arc that the angle subtends on a unit circle (a circle with radius 1). A radian is the angle created when you take the radius and lay it along the circle's edge. Hence, one full rotation around a circle equals \(2\pi\) radians because the circle's circumference is \(2\pi r\), where \(r = 1\) for a unit circle.
To convert between degrees and radians, you can use the relationship:

• 180 degrees = \(\pi\) radians.
• So, 1 degree = \(\frac{\pi}{180}\) radians.
• Conversely, 1 radian = \(\frac{180}{\pi}\) degrees.

For example, an angle of \(\pi/2\) radians is equivalent to 90 degrees. This understanding of radians is crucial when moving along the unit circle to locate terminal points.

Terminal Points

Terminal points are the final coordinates on the unit circle reached after moving a distance \(t\) radians starting from the point \((1, 0)\).
To find a terminal point:

• Start at \((1, 0)\), the initial point on the unit circle.
• Move counterclockwise if \(t\) is positive or clockwise if \(t\) is negative.
• The coordinates where you stop depend on the angle \(t\) radians.

Let's look at a few examples:
* For \(t = \pi/2\), you're moving 90 degrees counterclockwise, landing at \((0, 1)\).
* For \(t = \pi\), you move 180 degrees to the leftmost side of the circle, arriving at \((-1, 0)\).
* For \(t = -\pi/2\), go 90 degrees clockwise, reaching \((0, -1)\).
* For \(t = 2\pi\), you've completed a full 360-degree turn, thus returning to \((1, 0)\).
This process is a practical application of using radian measures to navigate the unit circle.

Coordinate Plane

The coordinate plane is a two-dimensional space defined by an x-axis (horizontal) and y-axis (vertical), which intersect at the origin \((0, 0)\). This plane is essential for visualizing the unit circle and plotting points based on calculations involving radian measures and angles.
Each point on the unit circle is described using coordinates \((x, y)\), which lies on the circle's circumference. Since the circle's center is at the origin:

• The unit circle equation is \(x^2 + y^2 = 1\).
• Coordinates are calculated using trigonometric functions: \(x = \cos(t)\) and \(y = \sin(t)\).

For example, at \(t = 0\), the terminal point is \((1, 0)\), since \(\cos(0) = 1\) and \(\sin(0) = 0\). At \(t = \pi/2\), the point is \((0, 1)\), with \(\cos(\pi/2) = 0\) and \(\sin(\pi/2) = 1\). Understanding how to use the coordinate plane with the unit circle is vital for finding terminal points. It allows for a clear and precise way to visualize and calculate angles and positions.