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Understanding Cosine with Similar Triangles

Understanding Cosine with Similar Triangles
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2017-06-26 02:53
TriangleVectorTrigonometryGeometric Construction
This example explains the concept of cosine using similar triangles. It involves the core mathematical concept of trigonometric functions, specifically cosine. Using commands from the construction summary, we see the use of points, segments, angles, and vectors. Suitable for students in middle school and above learning trigonometry.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(Ray[A, A + (0, 1)]) → C
  3. Segment(O, A) → adj
  4. Segment(O, C) → hyp
  5. Segment(C, A) → opp
  6. Point(adj) → B
  7. Segment(O, B) → b
  8. Angle(A, O, C) → α
  9. Segment((cos(α + 90°), sin(α + 90°)), ((3 * cos(α + 90°)), (3 * sin(α + 90°)))) → tick2
  10. Vector(((2 * cos(α + 90°)), (2 * sin(α + 90°))), ((2 * cos(α + 90°)), (2 * sin(α + 90°))) + ((hyp * cos(α)), (hyp * sin(α)))) → u
  11. Translate(tick2, u) → tick1
  12. Segment(O, (x(B), (opp / adj * x(B)))) → l
  13. Text(hyp, Point[((3 * cos(α + 90°)), (3 * sin(α + 90°))), Vector[u / 2]]) → text1
  14. Vector(O - (0, 2), A - (0, 2)) → v
  15. Segment(O - (0, 1), O - (0, 3)) → tick3
  16. Segment(A - (0, 1), A - (0, 3)) → tick4
  17. Segment(B, (x(B), (opp / adj * x(B)))) → h
  18. Text(adj, (adj / 2, -3)) → text2

AI Drawing Prompt Reference

Draw a right triangle with the right angle at point O, one leg OA as the adjacent side with length 1, and the other leg OC as the hypotenuse with length 3. At point C, draw an angle α equal to ∠AOC. From point C, draw a vector of length 3 in the direction of OC. From point A, draw a vector of length 1 in the direction of OA. Draw the unit vector corresponding to the cosine of angle α and translate it to point C. Draw the ratio of the adjacent side to the hypotenuse, which is cos(α), and label the hypotenuse as 3 and the adjacent side as 1.
The prompt above was compiled from the original drawing commands and is for reference only.