2. Trigonometric Ratios in the Unit Circle

The unit circle has a radius of 1 and is centered at the origin (0, 0).

The unit circle is a circle with a radius of 1. It is positioned on a coordinate plane such that its center is at the point (0, 0), which is known as the origin. This means that no matter where you are on the circle, the distance from the center to any point on the circle is always equal to 1. The unit circle is a fundamental concept in trigonometry because it allows us to define the sine and cosine functions for all angles, not just those that can be solved with right-angled triangles.

Think of the unit circle as a clock face where the center represents the time (0:00 or midnight). Every hour points to a different angle that represents a specific position on the circle, but the distance from the center (midnight) to any hour is always the same, just like the radius of the unit circle is always 1.

For any angle θ:

• cos(θ) = x-coordinate of point on unit circle
• sin(θ) = y-coordinate of point on unit circle.

In the context of the unit circle, for any angle θ measured from the positive x-axis, we can identify a corresponding point on the circle. The x-coordinate of this point gives us the cosine of the angle, while the y-coordinate gives us the sine. This means that for every angle, you can find its cosine and sine values just by looking at where it intersects the unit circle. For example, if θ is 30°, cos(30°) is √3/2 and sin(30°) is 1/2, corresponding to the coordinates of the point on the unit circle.

Imagine throwing a ball up into the sky at different angles. The height of the ball corresponds to the sine (y-coordinate) of the angle, while how far it has gone horizontally corresponds to the cosine (x-coordinate). Regardless of where the ball travels, its position can be mapped to a point on the unit circle.

Periodicity and symmetry:

• sin(θ + 360°) = sin(θ), cos(θ + 360°) = cos(θ)
• sin(−θ) = −sin(θ), cos(−θ) = cos(θ).

Trigonometric functions are periodic, which means their values repeat at regular intervals. For the sine and cosine functions, this interval is 360°. This is because when you complete one full rotation around the unit circle (360°), you return to the same coordinates, resulting in the same sine and cosine values. Additionally, these functions exhibit symmetry: the sine function is odd, meaning sin(−θ) is the negative of sin(θ), while cosine is even, meaning cos(−θ) equals cos(θ). This property can help simplify calculations involving these functions.

Consider the motion of a Ferris wheel. As it completes one full rotation, each point on the wheel returns to the same position, similar to how sine and cosine functions repeat their values every 360 degrees. If you flip a circle upside down, the horizontal positions will stay the same (cosine function), but the vertical positions will be negative (sine function).

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