Lesson 28 Working with Special Triangles PreCalculus 12252021

Lesson 28 – Working with Special Triangles Pre-Calculus 12/25/2021 Pre-Calculus 1

Review – Where We’ve Been n We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane n We also have an understanding of how to calculate the trig ratios of the angles in standard position 12/25/2021 Pre-Calculus 2

Lesson Objectives n Know the trig ratios of all multiples of 30°, 45°, 60°, 90° angles n Understand the concepts behind the trig ratios of special angles in all four quadrants n Solve simple trig equations involving special trig ratios n Tabulate the trig ratios to begin graphing trig functions 12/25/2021 Pre-Calculus 3

(A) Review – Special Triangles n n n n Review 45°- 90° triangle sin(45°) = sin(π/4) = cos(45°) = cos(π/4) = tan(45°) = tan(π/4) = csc(45°) = csc(π/4) = sec(45°) = sec(π/4) = cot(45°) = cot(π/4) = 12/25/2021 Pre-Calculus 4

(A) Review – Special Triangles n Review 45 -45 -90 triangle 12/25/2021 Pre-Calculus 5

(A) Review – Special Triangles n n n n Review 30°- 60°- 90° triangle 30° π/6 rad sin(30°) = sin(π/6) = cos(30°) = cos(π/6) = tan(30°) = cot(π/6) = csc(30°) = csc(π/6) = sec(30°) = sec(π/6) = cot(30°) = cot(π/6) = 12/25/2021 n Review 30°- 60°- 90° triangle 60° π/3 rad n sin(60°) = sin(π/3) = cos(60°) = cos(π/3) = tan(60°) = tan(π/3) = csc(60°) = csc(π/3) = sec(60°) = sec(π/3) = cot(60°) = cot(π/3) = n n n Pre-Calculus 6

(A) Review – Special Triangles n 30 -60 -90 triangle 12/25/2021 Pre-Calculus 7

(B) Trig Ratios of First Quadrant Angles n We have already reviewed first quadrant angles in that we have discussed the sine and cosine (as well as other ratios) of 30°, 45°, and 60° angles n What about the quadrantal angles of 0 ° and 90°? 12/25/2021 n Pre-Calculus 8

(B) Trig Ratios of First Quadrant Angles – Quadrantal Angles n n n n Let’s go back to the x, y, r definitions of sine and cosine ratios and use ordered pairs of angles whose terminal arms lie on the positive x axis (0° angle) and the positive y axis (90° angle) n sin(0°) = cos (0°) = tan(0°) = sin(90°) = sin(π/2) = cos(90°) = cos(π/2) = tan(90°) = tan(π/2) = 12/25/2021 Pre-Calculus 9

(B) Trig Ratios of First Quadrant Angles – Quadrantal Angles n n n n Let’s go back to the x, y, r definitions of sine and cosine ratios and use ordered pairs of angles whose terminal arms lie on the positive x axis (0° angle) and the positive y axis (90° angle) n sin(0°) = 0/1 = 0 cos (0°) = 1/1 = 1 tan(0°) = 0/1 = 0 sin(90°) = sin(π/2) =1/1 = 1 cos(90°) = cos(π/2) =0/1 = 0 tan(90°) = tan(π/2) =1/0 = undefined 12/25/2021 Pre-Calculus 10

(B) Trig Ratios of First Quadrant Angles - Summary n 12/25/2021 Pre-Calculus 11

(C) Trig Ratios of Second Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 120° (2π/3), 135° (3π/4) , 150° (5π/6) and 180° (π) 12/25/2021 Pre-Calculus 12

(C) Trig Ratios of Second Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 120°, 135°, 150° and 180° θ Sin(θ) Cos(θ) Tan(θ) 120° (2π/3) 150° (5π/6) 12/25/2021 Pre-Calculus 13

(C) Trig Ratios of Second Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 120°, 135°, 150° and 180° θ Sin(θ) Cos(θ) Tan(θ) 135° (3π/4) 180° (π) 12/25/2021 Pre-Calculus 14

(D) Trig Ratios of Third Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 210° (7π/6), 225° (5π/4), 240° (4π/3) and 270° (3π/2) θ Sin(θ) Cos(θ) Tan(θ) 210° (7π/6) 240° (4π/3) 12/25/2021 Pre-Calculus 15

(D) Trig Ratios of Third Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 210° (7π/6), 225° (5π/4), 240° (4π/3) and 270° (3π/2) θ Sin(θ) Cos(θ) Tan(θ) 225° (5π/4) 270° (3π/2) 12/25/2021 Pre-Calculus 16

(D) Trig Ratios of Fourth Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 300° (5π/3), 315° (7π/4), 330° (11π/4) and 360° (2π) θ Sin(θ) Cos(θ) Tan(θ) 300° (5π/3) 330° (11π/6 ) 12/25/2021 Pre-Calculus 17

(D) Trig Ratios of Fourth Quadrant Angles n Now let’s apply the same ideas & concepts to considering special second quadrant angles of 300° (5π/3), 315° (7π/4), 330° (11π/4) and 360° (2π) θ Sin(θ) Cos(θ) Tan(θ) 315° (7π/4) 360° (2π) 12/25/2021 Pre-Calculus 18

(G) Summary (As a Table of Values) 0 30 45 60 90 120 135 150 180 sin cos n 210 225 240 270 300 315 330 360 sin cos 12/25/2021 Pre-Calculus 19

(G) Summary – As a “Unit Circle” n The Unit Circle is a tool used in understanding sines and cosines of angles found in right triangles. n It is so named because its radius is exactly one unit in length, usually just called "one". n The circle's center is at the origin, and its circumference comprises the set of all points that are exactly one unit from the origin while lying in the plane. 12/25/2021 Pre-Calculus 20

(G) Summary – As a “Unit Circle” n 12/25/2021 Pre-Calculus 21

(H) Examples n Complete the worksheet: n http: //www. edhelper. com/math/trigonometry 1 04. htm n http: //www. edhelper. com/math/trigonometry 1 08. htm 12/25/2021 Pre-Calculus 22

(H) Trig Equations n Simplify the following: 12/25/2021 Pre-Calculus 23

(H) Trig Equations n Simplify or solve 12/25/2021 Pre-Calculus 24