3.03 - The Unit Circle - All Ratios

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

2024
Views:

182

Pages:

18
Author:

1 1

Trigonometry Lesson 3.03 – The Unit Circle and Remaining Ratios Tangent sin 𝜃 tan 𝜃 = cos 𝜃 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 𝜋 90° = 2 (0,1) The 90°s tan 𝜃 = 0° = 0 360° = 2𝜋 (1,0) 180° = 𝜋 (-1,0) sin 𝜃 tan 𝜃 = cos 𝜃 tan 𝜋 2 1 0 = ⇒ 𝐷𝑁𝐸 tan 𝜋 = 3𝜋 270° = 2 (0,-1) tan 3𝜋 2 𝑦 𝑥 = 0 −1 −1 0 =0 ⇒ 𝐷𝑁𝐸 0 1 tan 2𝜋 = = 0 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 sin 𝜃 tan 𝜃 = cos 𝜃 The 30°s tan 𝜃 = 1 𝜋 tan = 2 = 6 3 2 1 2 = 2 3 1 3 = 1 3 3 = 3 3 3 𝑦 𝑥 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 The 30°s sin 𝜃 tan 𝜃 = cos 𝜃 tan 𝜃 = 5𝜋 3 tan =− 6 3 𝜋 3 tan = 6 3 7𝜋 3 tan = 6 3 11𝜋 3 tan =− 6 3 𝑦 𝑥 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 The 45°s sin 𝜃 tan 𝜃 = cos 𝜃 tan 𝜃 = 2 𝜋 tan = 2 = 4 2 2 2 2 2 2 =1 𝑦 𝑥 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 The 45°s sin 𝜃 tan 𝜃 = cos 𝜃 tan 𝜃 = 3𝜋 tan = −1 4 𝜋 tan =1 4 5𝜋 tan =1 4 7𝜋 tan = −1 4 𝑦 𝑥 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 The 60°s sin 𝜃 tan 𝜃 = cos 𝜃 tan 𝜃 = 3 𝜋 tan = 2 = 1 3 2 3 2 2 1 = 3 𝑦 𝑥 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 The 60°s sin 𝜃 tan 𝜃 = cos 𝜃 tan 𝜃 = 2𝜋 tan =− 3 3 4𝜋 tan = 3 3 𝜋 tan = 3 3 5𝜋 tan =− 3 3 𝑦 𝑥 Secant 1 sec 𝜃 = cos 𝜃 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 90° = (0,1) 𝜋 2 The 90°s sec 𝜃 = 0° = 0 360° = 2𝜋 (1,0) 180° = 𝜋 (-1,0) 3𝜋 270° = 2 (0,-1) 1 sec 𝜃 = cos 𝜃 sec 𝜋 2 1 0 = ⇒ 𝐷𝑁𝐸 sec 𝜋 = 1 −1 3𝜋 2 1 0 sec 1 𝑥 = −1 = ⇒ 𝐷𝑁𝐸 1 1 sec 2𝜋 = = 1 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 1 sec 𝜃 = cos 𝜃 The 30°s sec 𝜃 = 𝜋 1 sec = = 1 6 3 2 = 2 3 2 3 = 2 3 2 3 = 3 3 3 1 𝑥 Cosecant 1 csc 𝜃 = sin 𝜃 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 90° = (0,1) 𝜋 2 The 90°s csc 𝜃 = 0° = 0 360° = 2𝜋 (1,0) 180° = 𝜋 (-1,0) 1 csc 𝜃 = sin 𝜃 csc 𝜋 2 1 𝑦 1 1 = =1 1 0 csc 𝜋 = ⇒ 𝐷𝑁𝐸 3𝜋 270° = 2 (0,-1) 3𝜋 1 csc = = −1 2 −1 1 0 csc 2𝜋 = ⇒ 𝐷𝑁𝐸 Cotangent cos 𝜃 cot 𝜃 = sin 𝜃 𝑥, 𝑦 → cos 𝜃 , sin 𝜃 90° = (0,1) 𝜋 2 The 90°s cot 𝜃 = 0° = 0 360° = 2𝜋 (1,0) 180° = 𝜋 (-1,0) cos 𝜃 cot 𝜃 = sin 𝜃 cot 𝜋 2 cot 𝜋 = 3𝜋 270° = 2 (0,-1) 𝑥 𝑦 0 1 = =0 −1 0 ⇒ 𝐷𝑁𝐸 3𝜋 0 cot = =0 2 −1 1 0 cot 2𝜋 = ⇒ 𝐷𝑁𝐸 Homework O Find the Tangent, Secant, Cosecant and Cotangent Ratios for our special angles (0, π/6, π/4, π/3, π/2) O Complete your positive Unit Circle with the final values for these four trigonometric ratios (in a box, next to each point).