☰ Contents · Algebra

Rotating a point about the origin

Lessons 19 · 1 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
19

Rotating a point about the origin

Textbook: pp. 97–102
GoalUnderstand rotating a point of the unit circle about the origin; find the coordinates of the rotated point and the angles giving the same point.
New words
unit circle · birlik aylanarotation · burishcounterclockwise · soat mili yo‘nalishiga qarama-qarshifull turn · to‘la aylanish
Explanation

A circle with centre at the origin and radius 1 is called the unit circle. Rotating the point P(1, 0) through α radians means moving the point along the circle a distance |α|: counterclockwise if α > 0, clockwise if α < 0; for α = 0 the point stays. Important rotations: π/2 → (0, 1), π → (−1, 0), 3π/2 → (0, −1), 2π → (1, 0); −π/2 gives (0, −1). The unit circle has length 2π, so a full turn is 2π: rotating through α and through α + 2πk (k an integer) gives the same point. Hence every point of the circle corresponds to infinitely many angles: α₀ + 2πk. We simplify a large angle by removing whole multiples of 2π.

Worked examples
5π = π + 2 · 2π, so rotating through 5π gives the same point as π, namely (−1, 0) (two full turns and a half turn). −7π/2 = π/2 − 2 · 2π, so the point is (0, 1).
All angles leading to the point (0, −1): −π/2 + 2πk, k = 0, ±1, ±2, ... (this is the same set as 3π/2 + 2πk).
Class activity

“Walk around the circle”: a big unit circle is drawn on the floor; a student stands at P, the class calls an angle α (π/2, −π, 5π/2...), and the student walks in that direction and says where they arrive.

Practice
1
Which point results from rotating P(1, 0) through π/2?
2
Which point results from rotating P(1, 0) through 4π?
3
Which point results from rotating P(1, 0) through 7π/2?
4
Why do rotations through α and through α + 2π give the same point?