6.2 Angles in Standard Position In All Quadrants.notebook
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Unit 6 - Trigonometry6.2 Angles in All Quadrants
6.2 Angles in Standard Position In All Quadrants.notebook
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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
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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
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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
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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
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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
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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
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?
?
?
Outcomes 6.2 & 6.8 6.11
6.2 Angles in Standard Position In All Quadrants.notebook
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?
?
?
Outcomes 6.2 & 6.8 6.11
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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.
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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
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(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
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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
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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
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Outcomes 6.13 & 6.14
6060
60
45
45
30 45 60
30
60
Lets determine the primary trig ratios of the following degrees:
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Example 2) Determine the measure of all angles in standard position with a reference angle of 60o.
Outcomes 6.13 & 6.14
x
y
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x
y
Your turn, try this one...Determine the measure of all angles in standard position with a reference angle of 30o.
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Outcomes 6.13 & 6.14
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Outcomes 6.2, 6.3, 6.9 & 6.10
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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
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Outcomes 6.2, 6.3, 6.9 & 6.10
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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
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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
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Outcome 6.9
Outcome 6.12
6.2 Angles in Standard Position In All Quadrants.notebook
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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
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Review
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