2/27/2016Pre-Calculus1 Lesson 28 – Working with Special Triangles Pre-Calculus

Review – Where We’ve Been We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane We also have an understanding of how to calculate the trig ratios of the angles in standard position 2/27/2016 Pre-Calculus 2

2/27/2016 Pre-Calculus 3 Lesson Objectives Know the trig ratios of all multiples of 30°, 45°, 60°, 90° angles Understand the concepts behind the trig ratios of special angles in all four quadrants Solve simple trig equations involving special trig ratios Tabulate the trig ratios to begin graphing trig functions

2/27/2016 Pre-Calculus 4 (A) Review – Special Triangles Review 45°- 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) =

2/27/2016 Pre-Calculus 5 (A) Review – Special Triangles Review triangle

2/27/2016 Pre-Calculus 6 (A) Review – Special Triangles 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) = Review 30°- 60°- 90° triangle  60°  π/3 rad 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) =

2/27/2016 Pre-Calculus 7 (A) Review – Special Triangles triangle

2/27/2016 Pre-Calculus 8 (B) Trig Ratios of First Quadrant Angles 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 What about the quadrantal angles of 0 ° and 90°?

2/27/2016 Pre-Calculus 9 (B) Trig Ratios of First Quadrant Angles – Quadrantal Angles 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) sin(0 °) = cos (0°) = tan(0°) = sin(90°) = sin (π/2) = cos(90°) = cos (π/2) = tan(90 °) = tan (π/2) =

2/27/2016 Pre-Calculus 10 (B) Trig Ratios of First Quadrant Angles – Quadrantal Angles 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) 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

2/27/2016 Pre-Calculus 11 (B) Trig Ratios of First Quadrant Angles - Summary

2/27/2016 Pre-Calculus 12 (C) Trig Ratios of Second Quadrant Angles 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)

2/27/2016 Pre-Calculus 13 (C) Trig Ratios of Second Quadrant Angles 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)

2/27/2016 Pre-Calculus 14 (C) Trig Ratios of Second Quadrant Angles 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)

2/27/2016 Pre-Calculus 15 (D) Trig Ratios of Third Quadrant Angles 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)

2/27/2016 Pre-Calculus 16 (D) Trig Ratios of Third Quadrant Angles 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)

2/27/2016 Pre-Calculus 17 (D) Trig Ratios of Fourth Quadrant Angles 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 )

2/27/2016 Pre-Calculus 18 (D) Trig Ratios of Fourth Quadrant Angles 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)

2/27/2016 Pre-Calculus 19 (G) Summary (As a Table of Values) sin cos sin cos

2/27/2016 Pre-Calculus 20 (G) Summary – As a “Unit Circle” The Unit Circle is a tool used in understanding sines and cosines of angles found in right triangles. It is so named because its radius is exactly one unit in length, usually just called "one". 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.

2/27/2016 Pre-Calculus 21 (G) Summary – As a “Unit Circle”

2/27/2016 Pre-Calculus 22 (H) Examples Complete the worksheet: 04.htm 04.htm 08.htm 08.htm

(H) Trig Equations Simplify the following: 2/27/2016 Pre-Calculus 23

2/27/2016 Pre-Calculus 24 (H) Trig Equations Simplify or solve