Mathematics

Question

In this discussion, you will discuss the relationships of key points on the unit circle. For the key points shown in the four quadrants on the unit circle, identify a significant pattern or symmetry in the angles, the coordinates, the trigonometric functions, or a relationship between them. Post a detailed response to the discussion prompt.

1 Answer

  • There are four key points on the unit circle.

    They are:

    1) Two iintersections with the x-axis
    2) Two intersections with the y-axis

    Intersections with the x-axis

    a) One of the point is 1 unit to the right of the origin (0,0). Then the intersections point is (1,0).

    Those  coordintatesidenfity the vector (1,0) whose angle is 0°.

    And the trigonometric functions sin, cos, and tan are:

    sin (0) = y-coordinate / radius of the circle = 0/1 =0

    cos(0) = x-coordinate / radius = 1/1 =1

    tan (0) = y-coordinate / x-coordinate = 0/1 = 0

    b) The other intersection point with the x-axis is one unit to the left of the center => (-1,0), and agle = 180°

    That drives to:

    sin(180°) = y-coordinate / radius = 0/(-1) = 0
    cos(180°) = x-coordinate / radius = -1/1 = -1
    tan(180°) = y-coordinate / x-coordinate = 0 /(-1) = 0

    Intersections with the y-axis

    c) One point is 1 unit up of the center => coordinates are (0,1) and angle is 90°

    Then,

    sin (90°) = y-coordinate/radius = 1/1 = 1
    cos(90°) = x-coordinate/ radius = 0/1 = 0
    tan(90°) = y-coordinate/x-coordinate = 1/0 = undefined


    d) The other intersection point with the y-axis is (-1,0), and the angle is 270°.

    Then:

    sin(270°) = y-coordinate / radius = -1/1 = -1
    cos(270°) = x-coordinate / radius = 0/1 = 0
    tan(270°) = y-coordinate / radius = -1/0 = undefined
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