Embed Size (px)
Citation preview
6.2 Angles in Standard Position In All Quadrants.notebook
1
May 24, 2015
Unit 6 - Trigonometry6.2 Angles in All Quadrants
6.2 Angles in Standard Position In All Quadrants.notebook
2
May 24, 2015
6.2 Angles in Standard Position in all Quadrants
The reference angle for an angle in standard position is the angle between___________________________________.
Thus, there are four different rotation angles (measured from the positive x-axis) that have the same reference angle.
As a result, any angle from 90 to 360 is the reflection the x-axis and/or y-axis of the angle's reference angle.
o oOutcome 6.4
the terminal arm and the x-axis
Outcome 6.3
x
y
Explain how to determine the angles from 0 to 360 that have the same reference angle as a given angle.
o o
6.2 Angles in Standard Position In All Quadrants.notebook
3
May 24, 2015
6.2 Angles in Standard Position in all Quadrants
For example, each of the following rotation angles have a reference angle of 35o but why? How is this possible?
Outcome 6.3 Explain how to determine the angles from 0 to 360 that have the same reference angle as a given angle.
o o
6.2 Angles in Standard Position In All Quadrants.notebook
4
May 24, 2015
Determine the reference angle for an angle in standard position. Sketch both angles.For an angle in Quadrant 2, such as 155o, the reference angle is:
For an angle in Quadrant 3, such as 235o, the reference angle is:
For an angle in Quadrant 4, such as 310o, the reference angle is:
Outcomes 6.1 & 6.2
6.2 Angles in Standard Position In All Quadrants.notebook
5
May 24, 2015
1
A 42
B 138
C 132
D 228
Determine the angle in standard position in the second quadrant that has a reference angle of 48.
6.2 Angles in Standard Position In All Quadrants.notebook
6
May 24, 2015
Now we need to find expressions for the three primary trigonometric ratios for angles in each of the four quadrants similarly to the way we did in Quadrant I:
Outcomes 6.8 6.10 Determine the value of sin θ, cos θ or tan θ, given any point P (x, y) on
the terminal arm of angle θ. Determine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0º, 90º, 180º, 270º or 360º.
Determine, using the Pythagorean theorem or the distance formula, the distance from the origin to a point P (x, y) on the terminal arm of an angle.
6.2 Angles in Standard Position In All Quadrants.notebook
7
May 24, 2015
The CAST Rule tells us where each of the primary trigonometric ratios are positive or negative. If a trigonometric ratio is not listed in a quadrant, it is negative in that quadrant.
C
AST
AllSin
Tan Cos
Determine the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ. Determine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0º, 90º, 180º, 270º or 360º.
Determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explain.
Outcome 6.9
Outcome 6.10
Outcome 6.11
6.2 Angles in Standard Position In All Quadrants.notebook
8
May 24, 2015
?
?
?
Outcomes 6.2 & 6.8 6.11
6.2 Angles in Standard Position In All Quadrants.notebook
9
May 24, 2015
?
?
?
Outcomes 6.2 & 6.8 6.11
6.2 Angles in Standard Position In All Quadrants.notebook
10
May 24, 2015
Each angle θ is in standard position. State the quadrants in which the terminal arm of the angle could lie.
a) sin θ = 4/5 b) cos θ = -1/2
c) tan θ = 1/5 d) sin θ = -0.25
Outcome 6.5 Determine the quadrant in which a given angle in standard position terminates.
6.2 Angles in Standard Position In All Quadrants.notebook
11
May 24, 2015
Example 1) The point B(2, -5) is on the terminal arm of an angle θ in standard position.a) Sketch the angle.
b) Determine the distance from the origin to point B. What is the name of this line? ___________________
c) Write the primary trigonometric ratios of θ.
d) Determine the measure of θ to the nearest degree.
Outcome 6.6
Outcome 6.8
Outcome 6.9
Outcome 6.12
6.2 Angles in Standard Position In All Quadrants.notebook
12
May 24, 2015
(a) R (radius)
(b) The reference angle
(c) θ Angle in standard position
(d) A Coterminal angle
(e) the General Solution
1b. A terminal arm is rotated through an angle θ in standard position. The terminal arm contains the point P(4,8). To one decimal place, find the value of:
x
y
Outcome 6.6
Outcome 6.8
Outcomes 6.3 & 6.4
Outcomes 6.9 & 6.12
Outcome 6.2
Outcome 6.12
6.2 Angles in Standard Position In All Quadrants.notebook
13
May 24, 2015
Your Turn, Try this one...The point B(-2, -4) is on the terminal arm of an angle θ in standard position.a) Sketch the angle.
b) Determine the distance from the origin to point B.
c) Write the primary trigonometric ratios of θ.
d) Determine the measure of θ to the nearest degree.
Outcome 6.8
Outcome 6.6
Outcome 6.12
Outcome 6.9
ç ü
6.2 Angles in Standard Position In All Quadrants.notebook
14
May 24, 2015
Consider these special triangles with a hypotenuse length of 1 unit. What are the other two side lengths?
1
300 45o
1
60o
1
Special TrianglesOutcomes 6.13 & 6.14
60o
1
6.2 Angles in Standard Position In All Quadrants.notebook
15
May 24, 2015
Outcomes 6.13 & 6.14
6060
60
45
45
30 45 60
30
60
Lets determine the primary trig ratios of the following degrees:
6.2 Angles in Standard Position In All Quadrants.notebook
16
May 24, 2015
Example 2) Determine the measure of all angles in standard position with a reference angle of 60o.
Outcomes 6.13 & 6.14
x
y
6.2 Angles in Standard Position In All Quadrants.notebook
17
May 24, 2015
x
y
Your turn, try this one...Determine the measure of all angles in standard position with a reference angle of 30o.
ç ü
Outcomes 6.13 & 6.14
6.2 Angles in Standard Position In All Quadrants.notebook
18
May 24, 2015
Outcomes 6.2, 6.3, 6.9 & 6.10
6.2 Angles in Standard Position In All Quadrants.notebook
19
May 24, 2015
Consider a rotation angle of 0o. In this case, x = 1, y = 0 and r = 1.
Consider a rotation angle of 90o. In this case, x = 0, y = 1, and r = 1.
Outcomes 6.9 & 6.10
6.2 Angles in Standard Position In All Quadrants.notebook
20
May 24, 2015
Outcomes 6.2, 6.3, 6.9 & 6.10
6.2 Angles in Standard Position In All Quadrants.notebook
21
May 24, 2015
Example 3 a) Given that , determine the other primary trigonometric ratios of the angle θ.
b) To the nearest degree, determine the possible values for θ when
Outcome 6.9
Outcome 6.12
6.2 Angles in Standard Position In All Quadrants.notebook
22
May 24, 2015
Your turn, try this one...a) Given that , determine the other primary trigonometric ratios of the angle θ.
b) To the nearest degree, determine the possible values for θ when
ç ü
Outcome 6.9
Outcome 6.12
6.2 Angles in Standard Position In All Quadrants.notebook
23
May 24, 2015
Example 4) a) Given that , determine the other primary trigonometric ratios of the angle α.
b) To the nearest degree, determine the possible values for α when
Outcome 6.9
Outcome 6.12
6.2 Angles in Standard Position In All Quadrants.notebook
24
May 24, 2015
Review
ç ü
ç ü
