Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Let \(s\) be a real number corresponding to the point ( \(a, b\) ) on the unit circle. Use this information to determine an expression representing the sine and cosine of each real number. $$s+\pi$$

Step-by-Step Solution

Answer

\(\cos(s+\pi) = -\cos(s)\) and \(\sin(s+\pi) = -\sin(s)\).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin. Any point \((a, b)\) on the unit circle satisfies the equation \(a^2 + b^2 = 1\). Here, the \(x\)-coordinate \(a\) of the point \((a, b)\) represents \(\cos(s)\) and the \(y\)-coordinate \(b\) represents \(\sin(s)\).

Step 2: Recognize the Angle Shift

Adding \(\pi\) to the angle \(s\) corresponds to rotating the point \((a, b)\) by 180 degrees around the origin. This transformation maps the point \((a, b)\) to \((-a, -b)\) on the unit circle.

Step 3: Determine the New Coordinates

The new coordinates after the transformation will be \((-a, -b)\). Thus, \(\cos(s + \pi) = -a\) and \(\sin(s + \pi) = -b\).

Step 4: Write the Expressions for Sine and Cosine

Based on the new coordinates, the expressions representing the sine and cosine of \(s + \pi\) are: \(\cos(s + \pi) = -\cos(s)\) and \(\sin(s + \pi) = -\sin(s)\).

Key Concepts

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that arise when working with the unit circle. The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. On this circle, each point, corresponding to an angle, represents coordinates \(a, b\). In trigonometry, these coordinates are associated with the trigonometric functions as follows: the x-coordinate is the cosine of the angle, and the y-coordinate is the sine of the angle.

• \(\cos(s) = a\)
• \(\sin(s) = b\)

This connection between coordinates and angles allows us to use the unit circle for a geometric understanding of sine and cosine. The values of sine and cosine depend on the angle, and they help us to determine other trigonometric values, which are widely used in mathematics, physics, and engineering.

Angle Transformation

Angle transformation involves modifying an angle by adding or subtracting a certain amount, resulting in a new angle with potentially different sine and cosine values. In our exercise, adding \(\pi\) (180 degrees) to the angle \(s\) means rotating the point \((a, b)\) around the origin by half a turn on the unit circle. This rotates the point to \((-a, -b)\).
This transformation is crucial because it helps us understand the cyclical nature of the trigonometric functions. By transforming angles, we can predict changes to the values of sine and cosine without having to depend strictly on their original positions; instead, we follow their periodic properties:

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

These transformations illustrate the symmetry of the trigonometric functions over their periods and are essential for understanding graphs and patterns in trigonometry.

Coordinate Geometry

Coordinate geometry combines algebra and geometry to discuss points, lines, and figures on a graph using a coordinate system. In the context of the unit circle and trigonometry, it helps in analyzing movements and transformations of points.
When using coordinate geometry, each point on the unit circle can be described using coordinates that adhere to the equation \(a^2 + b^2 = 1\). This makes it intuitive to relate geometric transformations to changes in trigonometric functions.
Specifically, when the angle \(s\) is adjusted, the resulting coordinates shift. Altering \((a, b)\) to \((-a, -b)\) by adding \(\pi\) highlights the relationship between coordinates and trigonometric function values.

• Transformations such as these manifest changes in symmetry.
• The transition from one set of coordinates to another provides insights into angles and rotations.

By using coordinate geometry, it becomes easier to visually and algebraically interpret how points on the unit circle reflect variations in angles and effectively predict corresponding sine and cosine values.