3.02 - The Unit Circle - Definitions, Coordinates

University:
High School
Course:
High School Physics
Academic year:

2024
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141

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20
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Trigonometry 3.02 – The Unit Circle, Definitions, Coordinates Centered at the Origin The Unit Circle (0,0) Radius is 1 unit Circumference 𝐶 = 2𝜋𝑟 = 2𝜋(1) = 2𝜋 Arms, Rotations, Standard Position “Positive” rotation = counter-clockwise “Negative” rotation = clockwise (0,0) Initial Arm 0° (1,0) never moves Terminal Arm The final rotation marker Standard position = any angle with initial arm on positive x-axis Principal Angle 0° A positive rotation between 0 - 360° (or 0 - 2π rad) is called a “principal angle” Related Acute Angle 0° For quads 2,3,4, the acute angle made between terminal arm and x-axis is called “related acute angle” Coordinates For our introduction to the unit circle, we will begin with the POSITIVE (Counter-Clockwise) rotation The 90°s 90° (0,1) 0° (1,0) 180° (-1,0) 270° (0,-1) The 45°s 135° 45° 0° 225° 315° Pythagorean 𝑥 2 + 𝑦 2 = 12 The 45°s 𝑥=𝑦 𝑥 2 + (𝑥)2 = 12 2𝑥 2 = 1 1u 1 2 𝑥 = 2 𝑥= 𝑥= 45° 135° 45° y 0° 45° 1 2 x 1 2 2 𝑥= =𝑦 2 225° 315° The 45°s − 2 2 , 2 2 135° 45° 45° 45° 2 2 225° − ,− 2 2 2 2 , 2 2 0° 45° 315° 2 2 ,− 2 2 The 30°s 150° 30° 0° 210° 330° 30-60-90 1 𝑦= 2 Pythagorean 2 1 𝑥2 + = 12 2 The 30°s 1 𝑥 + =1 4 2 𝑥2 = 𝑥= 𝑥= 𝑥= 3 4 3 4 3 4 3 2 1u y 30° x The 30°s 3 1 − , 2 2 150° 3 1 , 30° 2 2 30° 30° 210° 3 1 − ,− 2 2 0° 30° 330° 3 1 ,− 2 2 The 60°s 120° 60° 0° 240° 300° 30-60-90 1 𝑥= 2 Pythagorean 2 1 + 𝑦 2 = 12 2 The 60°s 1 + 𝑦2 = 1 4 𝑦2 = 𝑦= 3 4 3 4 30° y 1u 30° 60° x 𝑦= 𝑦= 3 4 3 2 The 60°s 1 3 − , 2 2 60° 120° 1 3 , 2 2 0° 60° 60° 60° 1 3 240° − ,− 2 2 300° 1 3 ,− 2 2 Ratios and their Angles Connection to the Coordinates 𝐴 𝑥 cos 𝜃 = = = 𝑥 𝐻 1 𝑂 𝑦 sin 𝜃 = = = 𝑦 𝐻 1 30° y 0° 30° x Coordinates 𝑥, 𝑦 (cos 𝜃 , sin 𝜃) Summary (quad I): O 0° 1,0 O 30° 3 1 , 2 2 O 45° 2 2 , 2 2 O 60° 1 3 , 2 2 O 90° (0, 1) AND 𝑥, 𝑦 = (cos 𝜃 , sin 𝜃) Numbers to Know 1 3 2 0, ±1, ± , ± ,± 2 2 2 Homework O Draw and fill in 2 Unit Circles: O (1) Positive Rotations with the coordinates O (2) Negative Rotations with the coordinates O **NOTE** This is to be a progressive work, so a FULL sheet for each circle is recommended