Problem 8 - Precalculus: Functions and Graphs

Question

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) \(P(-t-\pi)\) $$\left(\frac{7}{25},-\frac{24}{25}\right)$$

Step-by-Step Solution

Answer

(a) \((-\frac{7}{25}, \frac{24}{25})\), (b) \((-\frac{7}{25}, \frac{24}{25})\), (c) \((\frac{7}{25}, \frac{24}{25})\), (d) \((-\frac{7}{25}, -\frac{24}{25})\).

Step 1: Analyze the point on the unit circle

Recognize that the given point \(P(t)\) \(\left(\frac{7}{25}, -\frac{24}{25}\right)\) lies on the unit circle \(U\). The coordinates represent \((\cos t, \sin t)\). Therefore, \(\cos t = \frac{7}{25}\) and \(\sin t = -\frac{24}{25}\).

Step 2: Calculate \(P(t+\pi)\)

Using the properties of the unit circle, \(P(t+\pi)\) corresponds to \((-\cos t, -\sin t)\). Thus, \(P(t+\pi) = \left(-\frac{7}{25}, \frac{24}{25}\right)\).

Step 3: Calculate \(P(t-\pi)\)

Since \(P(t-\pi)\) is equivalent to \(P(t+\pi)\), the point is \((-\cos t, -\sin t)\). So, \(P(t-\pi) = \left(-\frac{7}{25}, \frac{24}{25}\right)\).

Step 4: Calculate \(P(-t)\)

For \(P(-t)\), the coordinates are \((\cos(-t), \sin(-t)) = (\cos t, -\sin t)\), using the even and odd properties of cosine and sine. Therefore, \(P(-t) = \left(\frac{7}{25}, \frac{24}{25}\right)\).

Step 5: Calculate \(P(-t-\pi)\)

Combine the transformations for \(P(-t)\) and \(P(t+\pi)\), you find \(P(-t-\pi) = (-\cos t, \sin t) = \left(-\frac{7}{25}, -\frac{24}{25}\right)\).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when working with the unit circle and solving problems involving angles and coordinates. Some key trigonometric identities include:

• Reciprocal identities: \(\csc t = \frac{1}{\sin t}\), \(\sec t = \frac{1}{\cos t}\), \(\cot t = \frac{1}{\tan t}\)
• Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\)
• Even and odd identities: \(\cos(-t) = \cos t\) and \(\sin(-t) = -\sin t\)

These identities help derive other useful formulas and transformations, such as those needed for evaluating the trigonometric functions at specific points on the unit circle.
For example, when asked to find \(P(t-\pi)\), knowing that \(\sin(t+\pi) = -\sin t\) and \(\cos(t+\pi) = -\cos t\) allows quick determination of the corresponding point.

Cosine and Sine

Cosine and sine are fundamental trigonometric functions that represent the rectangular coordinates of a point on the unit circle. When working with angles:

• \(\cos t\) represents the x-coordinate (horizontal component).
• \(\sin t\) represents the y-coordinate (vertical component).

These relationships offer a simple yet powerful way to interpret the unit circle.
The unit circle is a circle with a radius of one, centered at the origin \(0,0\) in the coordinate system. As a point moves around the circle, the coordinates change according to the angle \(t\), which can be measured in radians or degrees.
This understanding allows transformations like flipping signs or modifying angles while still maintaining the structure of the unit circle.

Coordinate Transformations

Coordinate transformations involve changing the position or orientation of a point within a coordinate system, such as the unit circle. These transformations can include rotations, reflections, or translations based on angle changes.
A common transformation involves translating the angle by \(\pi\) units, achieving points like \(P(t+\pi)\), which requires flipping both coordinates, \(\cos t\) and \(\sin t\), giving \(-\cos t\) and \(-\sin t\) accordingly.

• For reflection over the y-axis (e.g., \(P(-t)\)), the x-coordinate remains the same while the sine changes sign: \(\left(\cos t, -\sin t\right)\).
• For rotation by \(\pi\) radians (half-circle turn), both coordinates change signs: \(\left(-\cos t, -\sin t\right)\).

Understanding these transformations is essential for solving problems in trigonometry and aids in visualizing changes in points on the unit circle.

Rectangular Coordinates on Unit Circle

In trigonometry, the unit circle highlights the relationship between angles and their respective coordinate points. A significant part of this relationship is understanding rectangular coordinates, which describe positions using cosine and sine values.
Given a point on the unit circle corresponding to an angle \(t\), like \(P(t) = (\cos t, \sin t)\), these coordinates show the exact position on an x-y plane.

• The x-coordinate is derived from the cosine of the angle.
• The y-coordinate is derived from the sine of the angle.

These coordinates both describe a point on the unit circle where the equation \(\cos^2 t + \sin^2 t = 1\) always holds true, confirming the point lies on the circle.
This knowledge facilitates the navigation of points through different angle transformations and contributes greatly to solving trigonometric equations.