lesson 28 – working with special...
TRANSCRIPT
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3/3/14 Pre-Calculus 1
Lesson 28 – Working with Special Triangles
Pre-Calculus
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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
3/3/14 Pre-Calculus 2
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3/3/14 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
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3/3/14 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) =
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3/3/14 Pre-Calculus 5
(A) Review – Special Triangles
Review 45-45-90 triangle
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3/3/14 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) =
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3/3/14 Pre-Calculus 7
(A) Review – Special Triangles
30-60-90 triangle
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3/3/14 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°?
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3/3/14 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) =
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3/3/14 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
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3/3/14 Pre-Calculus 11
(B) Trig Ratios of First Quadrant Angles - Summary
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3/3/14 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)
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3/3/14 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)
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3/3/14 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)
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3/3/14 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)
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3/3/14 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)
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3/3/14 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)
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3/3/14 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)
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3/3/14 Pre-Calculus 19
(G) Summary (As a Table of Values)
0 30 45 60 90 120 135 150 180
sin
cos
210 225 240 270 300 315 330 360
sin
cos
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3/3/14 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.
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3/3/14 Pre-Calculus 21
(G) Summary – As a “Unit Circle”
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3/3/14 Pre-Calculus 22
(H) Examples
Complete the worksheet:
http://www.edhelper.com/math/trigonometry104.htm
http://www.edhelper.com/math/trigonometry108.htm
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(H) Trig Equations
Simplify the following:
3/3/14 Pre-Calculus 23
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3/3/14 Pre-Calculus 24
(H) Trig Equations
Simplify or solve