Benchmark Angles and Special Angles Math 30-11. Deriving the Equation of a Circle P(x, y) O (0, 0) Note: OP is the radius of the circle. The equation.

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Presentation transcript:

Benchmark Angles and Special Angles Math 30-11

Deriving the Equation of a Circle P(x, y) O (0, 0) Note: OP is the radius of the circle. The equation of a circle with its centre at the origin (0, 0) is x 2 + y 2 = r 2. Math 30-12

Determine the equation of a circle with centre at the origin and a radius of a) 2 unitsb) 5 units c) 1 unit A circle of radius 1 unit with centre at the origin is defined to be a Unit Circle. When r = 1, a = θr becomes a = θ. The central angle and its subtended arc on the unit circle have the same numerical value. Math 30-13

Coordinates on the unit circle P(x, y) satisfy the equation Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case. a) Why are there two answers? Point P(x,y) McGraw Hill Teacher Resource DVD 4.2_193_IA A point P(x, y) exists where the terminal arm intersects the unit circle. Math 30-14

Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case. b) The Points of intersection are in quadrants III and IV. c) The point is the point of intersection of a terminal arm and the unit circle. What is the length of the radius of the circle? A unit circle, by definition, has a radius of 1 unit. Math 30-15

Relating Arc Length and Angle Measure in Radians The function P(θ) = (x, y) can be used to relate the arc length, θ, of a central angle, in radians, in the unit circle to the coordinates, (x, y) of the point of intersection of the terminal arm and the unit circle. When θ = π, the point of intersection is (-1, 0), This can be written as P( π ) = (-1, 0) Determine the coordinates of the point of intersection of the terminal arm and the unit circle for each: (1, 0) (0, -1) Math 30-16

Special Triangles from Math Unit Circle with Right Triangle Present Math 30-17

Exploring Patterns for Reflect in the y-axis and in the x-axis Convert to a Radius of 1 Math 30-18

Exploring Patterns for Convert to a Radius of 1 Convert to a Radius of 1 Math 30-19

(1, 0) (0, 1) (-1, 0) (0, -1) The Unit Circle WUAOOLIXQSG4U_RISTHU

(1, 0) (0, 1) (-1, 0) (0, -1) The Unit Circle Math

Page 186 1a, 2a,b, 3a,d, 4a,b,c,d,g, 5a,c,g,I 10, 12, 15 Math