Transcript Lesson 6B: Angles in Standard Position

Math 20-1 Chapter 2 Trigonometry
2.1B Angles in Standard Position
Teacher Notes
Math 20-1 Chapter 1 Sequences and Series
2.1B Angles in Standard Position Exact Values
2.1.1
Chapter
Angles in Standard Position
Identify the angles sketched in standard position.
Check answer
2.1.2
Torso Angle - Fast
Torso angle is very dependent upon the
cyclists choice of performance and
comfort. A lower position is more
aerodynamic as frontal surface area is
reduced. 30° to 40° is a good
compromise of performance and
comfort but does rely on reasonably
good flexibility to lower back and
hamstrings.
Torso Angle - Touring
A more relaxed torso angle will take the
pressure off the lower back, hamstrings
and the neck and distribute loads from
hands to seat. 40° to 50° is a suitable
angle for longer distances where
comfort is the priority over speed.
2.1.3
Reference Angles
Determine the
measure of the
reference angle.
Angle in
Standard
Position (θ)
165°
320°
250°
60°
Reference
Angle (θR)
Quadrant
85°
46°
37°
52°
III
I
IV
II
Quadrant
Reference
Angle (θR)
II
IV
III
15°
40°
70°
I
60°
Angle in
Standard
Position (θ)
265°
46°
Determine the
measure of the
angle in standard
position.
323°
128°
2.1.4
A ship is sailing in a direction given by the bearing N35°E.
Sketch the angle.
35°
55°
What is the measure of the angle in standard position?
55°
What is the measure of the reference angle of the angle in
standard position?
55°
2.1.5
Trigonometry compares the ratios of the sides in a right triangle.
The Primary Trigonometric Ratios
There are three primary trig ratios:
Opposite the right-angle

sine
cosine
opp
adj
sin  
cos 
hyp
hyp
tangent
opp
tan  
adj
Next to the angle
1
2
30º
sin 30 
1
2
2.1.6
Trig Equations
sin 30º=
trig
function
300
angle
1
2 or 0.5
trig ratio
Knowing the measure of the
reference angle, can you label the
triangle?
2.1.7
Exact Values for Trig Ratios of Special Angles
300 - 600 - 900
600
2
2
c2 = a2 + b2
22 = a2 + 12
22 - 12 = a2
√3 = a
300
2
3
600
600
600
2
1
450 - 450 -900
1
2
450
c2 = a2 + b2
= 12 + 12
=2
c=√2
450
1
2.1.8
Exact Values of Trig Ratios
1
2
3
2
1
3
3
2
1
2
3
1
2
1
2
1
2.1.9
Angle
30°
150°
210°
330°
Quadrant Sin
1
I
2
1
II
2
III

1
2
IV

1
2
Cos
Tan
1
3
3
2

3
2

3
2
3
2
1
3

1
3

1
3
What do the angles have in common?
What do notice about the ratios of the lengths of sides?
Make a conjecture to determine the sign of the trig ratio for
each quadrant.
2.1.10
Use your
conjecture to
determine
the sign of
the trig ratio
for each
quadrant.
Angle
Quadrant Sin
60°
120°
240°
III
300°
IV
Angle
45°
135°
225°
315°
Cos
Tan
3
2
1
2
3
2

3
2
3

2

I
II
1
2
1
2

Quadrant Sin
1
I
2
1
II
2
III

1
2
IV

1
2
1
2
Cos
1
2
1

2

1
2
1
2
3
 3
3
 3
Tan
1
1
1
1
2.1.11
Allie is learning to play the piano. Her teacher
uses a metronome to help her keep time. The
pendulum arm of the metronome is 10 cm long.
For one particular tempo, the setting results in the
arm moving back and forth from a start position
of 60° to 120°. What is the exact horizontal
distance the tip of the arm moves in one beat?
Calculate the horizontal distance to the midline, labeled a.
a
10
Which trig ratio would you use to determine the length of side a?
1 a
adj
The exact horizontal

cos 60 
2
10
hyp
distance is 10 cm.
a
cos 60 
10
60°
a
10
a
2
5a
2.1.12
Bad Math Jokes:
Two angles meet at a dance.
How did Mr. 150° get Miss 30° to say yes to a dance?
We share the same sine, we
were made for each other.
Who should Mr. 300° ask to dance?
2.1.13
Using Exact Values Homework
State the value of each ratio.
300
1. sin
=
3. tan 450 =
5. sin
1500
=
RA = 300
7. tan 1350 =
RA = 450
9. sin
1350
RA = 450
=
1
2
2. cos
1
4. sin 600 =
1
2
1200
6. cos
450
1
2
3
2
=
=
RA = 600
1
1
2
8. tan 1200 =
RA = 600
10. cos
1500
=
RA = 300
1

2
 3
3

2
2.1.14
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8, 9, 13, 16, 17b, 24a,b
2.1.15