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Cosine Ratio: Trigonometric Function via Unit Circle and Right Triangle

Cosine Ratio: Trigonometric Function via Unit Circle and Right Triangle
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2014-12-18 13:05
CircleAngleTrigonometryGeometric ConstructionSlider Animation
This GeoGebra example demonstrates the cosine ratio using both a unit circle and a right triangle, visually connecting the geometric definition (adjacent over hypotenuse) with the coordinate on the unit circle. Core mathematical concepts include the cosine function, the unit circle, and the relationship between angles and x-coordinates. The construction uses commands such as Ray, Point, Segment, OrthogonalLine, Polygon, and Angle, with objects like B, B_1, A, C, D, poly1, and multiple text labels (text1 to text13) plus auxiliary objects ta, si, co to display side lengths, ratios, and function values. This resource is suitable for high school or introductory college students learning trigonometric definitions, unit circle properties, and dynamic exploration of how cosine changes with angle. It serves as an interactive teaching tool for instructors or a self-study aid for learners.

GeoGebra Commands & Parameters

  1. Ray(B, B_1) → a_1
  2. Point(a_1) → A
  3. Segment(A, B) → c
  4. OrthogonalLine(B, c) → c_2
  5. Point(c_2) → D
  6. Ray(B, D) → d
  7. Point(d) → C
  8. Polygon(C, A, B) → poly1, b, e, a
  9. Angle(B, A, C) → α
  10. Angle(A, C, B) → β
  11. Angle(C, B, A) → γ

AI Drawing Prompt Reference

Draw a unit circle and a right triangle to demonstrate the cosine ratio, showing the connection between the geometric definition (adjacent over hypotenuse) and the coordinate on the unit circle by sliding the angle. Start by drawing a ray, then place a point on it, and connect this point to the circle's center to form a segment. Next, draw a perpendicular line to this segment that intersects the circle at another point, creating a new ray from the center to this point. Connect the center to the intersection point of the new ray and the perpendicular line to form a right triangle. Label the three angles and display the angle values and cosine values, dynamically showing how the cosine changes with the angle.
The prompt above was compiled from the original drawing commands and is for reference only.