Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-8: Let \(P(t)\) be the point on the unit circle \(U\) that corresponds to \(t\). If \(P(t)\) has the given rectangular coordinates, find (a) \(P(t+\pi)\) (b) \(P(t-\pi)\) (c) \(P(-t)\) (d) \(P(-t-\pi)\) $$ \left(\frac{3}{5}, \frac{4}{5}\right) $$

Step-by-Step Solution

Answer

(a) \((-\frac{3}{5}, -\frac{4}{5})\), (b) \((-\frac{3}{5}, -\frac{4}{5})\), (c) \((\frac{3}{5}, -\frac{4}{5})\), (d) \((-\frac{3}{5}, \frac{4}{5})\).

Step 1: Understand the Unit Circle

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. Any point \(P(t)\) on the unit circle can be defined by coordinates \((\cos t, \sin t)\). Here, the given coordinates \(\left( \frac{3}{5}, \frac{4}{5} \right)\) represent \((\cos t, \sin t)\) on the unit circle.

Step 2: Determine 't' for Given Point

Identify \(t\) such that \(\cos t = \frac{3}{5}\) and \(\sin t = \frac{4}{5}\). This means \(t\) is an angle whose cosine is \(\frac{3}{5}\) and sine is \(\frac{4}{5}\).

Step 3: Find P(t+π)

On the unit circle, adding \(\pi\) to angle \(t\) gives the point \((\cos(t+\pi), \sin(t+\pi)) = (-\cos t, -\sin t)\). So, \(P(t+\pi) = \left(-\frac{3}{5}, -\frac{4}{5}\right)\).

Step 4: Find P(t-π)

Subtracting \(\pi\) from angle \(t\) yields the point \((\cos(t-\pi), \sin(t-\pi)) = (-\cos t, -\sin t)\). Therefore, \(P(t-\pi) = \left(-\frac{3}{5}, -\frac{4}{5}\right)\).

Step 5: Find P(-t)

The point opposite to \(t\) (i.e., \(-t\)) on the unit circle is \((\cos(-t), \sin(-t)) = (\cos t, -\sin t)\). Hence, \(P(-t) = \left(\frac{3}{5}, -\frac{4}{5}\right)\).

Step 6: Find P(-t-π)

Adding \(-\pi\) to \(-t\) gives \((\cos(-t-\pi), \sin(-t-\pi)) = (-\cos t, \sin t)\). Thus, \(P(-t-\pi) = \left(-\frac{3}{5}, \frac{4}{5}\right)\).

Key Concepts

Trigonometric Coordinates

Trigonometric coordinates are used to describe the position of a point on the unit circle. The unit circle is a circle centered at the origin (0,0) with a radius of 1. Each point on this circle can be represented by the coordinates

• ( \( \cos t \) , \( \sin t \) )

where the angle t is the measurement from the positive x-axis to the line segment that joins the origin to the point.
These coordinates are crucial in defining the trigonometric functions sine and cosine based on an angle.
In our example, the coordinates

• ( \( \frac{3}{5} \) , \( \frac{4}{5} \) )

represented a point on the unit circle meaning that

• \( \cos t = \frac{3}{5} \)
• \( \sin t = \frac{4}{5} \)

Understanding trigonometric coordinates helps you discover how angles relate to points on the circle.

Cosine and Sine

Cosine and sine are fundamental trigonometric functions that describe the x and y coordinates of a point on the unit circle (respectively).
For a given angle,

• \( \cos (t) \)
• \( \sin (t) \)

represents the horizontal and vertical components of the point on the circle.
In the unit circle, the cosine value gives how far left or right the point is from the center, while the sine value gives how far up or down the point is.When the point on the unit circle has the coordinates

• ( \( \frac{3}{5} \) , \( \frac{4}{5} \) )

in our example, cosine is

• \( \frac{3}{5} \)

and sine is

• \( \frac{4}{5} \)

. Understanding cosine and sine is essential in finding positions on the unit circle based on angles.

Angle Addition

Angle addition allows us to find new points on the unit circle by adding a specific angle to the current one. This is crucial in rotating points around the circle.
For example, adding \(\pi\) (or 180 degrees) to an angle moves a point to directly opposite on the circle.
This means

• \( \cos(t+\pi) = -\cos(t) \)
• \( \sin(t+\pi) = -\sin(t) \)

So, with the original point being

• ( \( \frac{3}{5} \) , \( \frac{4}{5} \) )

, adding \(\pi\) gives us the new point

• (-\( \frac{3}{5} \) , -\( \frac{4}{5} \) )

. This technique helps understand reflections of points over the origin in the coordinate plane.

Coordinate Transformation

Coordinate transformation is a way of mapping points from one coordinate space to another. For the unit circle, this involves changing the angle while observing how that changes the point's coordinates.
Transformations like

• \( t \to -t\)

, cause the point to flip across the x-axis, turning

• \( (\cos(t), \sin(t)) \)
into
• \( (\cos(t), -\sin(t)) \)
. Similarly, transforming by
• \(-t-\pi\)
, the coordinates become
• (-\cos(t), \sin(t))
. This technique aids in visualizing changes in position and orientation of points on the unit circle through precise transformations based on angle adjustments.