91Ó°ÊÓ

The unit circle is a vital concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. As you move around the unit circle, the angle formed with the positive x-axis defines a point on the circle. The coordinates of this point are determined by the corresponding angle, \( \theta \). On the unit circle, each angle is associated with a specific point whose coordinates are given by \((\cos \theta, \sin \theta)\).

• At \( \theta = 0 \), the point is \((1, 0)\).
• At \( \theta = \pi/2 \) or 90 degrees, the point is \((0, 1)\).
• At \( \theta = \pi \) or 180 degrees, the point is \((-1, 0)\).
• At \( \theta = 3\pi/2 \) or 270 degrees, the point is \((0, -1)\).

The unit circle allows us to visualize the repetitive nature of the trigonometric functions such as sine and cosine, as angles beyond \( 2\pi \) will correspond to points already on the circle, thus highlighting the periodic nature of these functions.

The cosine function is a fundamental component in trigonometry. It is often used to determine the horizontal position (or x-coordinate) of a point on the unit circle at a given angle \( \theta \).

• Cosine tells us how far along the x-axis a point lies relative to the unit circle's center.
• Its range is between -1 and 1, correlating to the extreme left and right points on the unit circle.

When \( \cos \theta = -1 \), as in the original exercise, it pinpointly describes the point on the unit circle where the angle \( \theta \) results in the x-coordinate being exactly -1. This occurs precisely at \( \theta = \pi \) radians (180 degrees) and indicates the farthest point left on the unit circle. This is a crucial angle as it represents a key "corner" in trigonometric analysis.

Trigonometric equations often feature periodic solutions due to the repeating nature of functions like cosine and sine. These functions reset every \( 2\pi \) radians as we circle around the unit circle.
To express all possible solutions for a trigonometric equation, we use a formula that incorporates this periodicity.
For the equation \( \cos \theta = -1 \), the solution is found initially at \( \theta = \pi \). However, due to periodicity, this solution repeats every \( 2\pi \) radians.

• The general solution can therefore be expressed as \( \theta = \pi + 2k\pi \).
• Here, \( k \) is any integer (positive, negative, or zero), signifying that as we add multiples of \( 2\pi \), we arrive at other solutions.

This form is essential for solving trigonometric equations as it accounts for every possible angle that can yield a particular cosine or sine value across continuous cycles of the circle.