SECTION 2.1ANGLES IN STANDARD POSITION
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I) MEASURING ANGLES IN DEGREES
When working with angles of rotation:
Counter clockwise: Positive Angle
Clockwise: Negative Angle
Begin on the right side at zero degrees
Quarter Circle:
Half Circle:
3 Quarters Circle
Full Circle:
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II) QUADRANTS & X /Y AXIS
Center: Origin (0,0)
On the X-axis:
Right – Positive
Left - Negative
On the Y-axis:
Up – Positive
Bottom - Negative
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III) UNIT CIRCLES & ROTATIONS
A “unit circle” is a circle rotated around the origin with a radius of1 unit
The radius starts to rotate from the right side (Initial Arm)
The line rotating around the center is called a “Terminal Arm”
Rotated Counter clock-wise (positive angle)
Rotated Clock-wise (negative angle)
PositiveDirection
NegativeDirection
Initial Arm
Terminal Arm
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IV) ANGLES IN STANDARD POSITION:
All angles in “standard position” begin from the Initial arm(right)
The angle is created by rotating the Terminal arm aroundthe origin (counter-clockwise)
in standard position
Ex: Draw the following angles in standard position: a) 62° b) 152 °
in standard position
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Ex: Draw the following angles in standard position:
a) 312° b) -77°
in standard position
in standard position
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c) 450° d) 540°
450° in standard position
540° radians instandard position
VI) CO-TERMINAL ANGLES
Angles that have their terminal arms at the same position
Co-terminal angles have a difference of 360° or multiples of360° (Full circles)
ie: 30°, 390°, 750°, -330°, -330°, -690°, ..etc
All these angles have the same
Position, “Co-terminal Angles”
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Ex: Given the following angles, provide two co-terminalangles and a formula for all the co-terminal angles
Add/subtract full circles
to get co-terminal angles
All co-terminal are sums/differences of 360°
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Add/subtract full circles
to get co-terminal angles
All co-terminal are sums/differences of 360°
VI) REFERENCE ANGLES
An angle created by the terminal arm and X-axis.
Reference angles must be in the same quadrant as theterminal arm
Ex: Given the following angles in standard position, find the
reference angle
in standard position
in standard position
ReferenceAngle
ReferenceAngle
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Ex: Given each of the following terminal arms, indicatewhich is the reference angle:
NEITHER!!
NOTE: THE Reference angles must be formed with the X-axis and it must be in the same quadrant as the terminal arm
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Practice: Find the reference angle for each of the followingangles in standard position:
ReferenceAngle
Reference Angle
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PRACTICE: FIND THE REFERENCE ANGLE FOR EACH OFTHE FOLLOWING ANGLES IN STANDARD POSITION
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VII) RECOGNIZING REFERENCE ANGLES
Angles with the same height will have the same reference angle
ie: 30°, 150°, 210°, and 330° all have the same reference angle at 30°
ie: 60°, 120°, 240°, and 300° all have the same reference angle at 60°
ie: 45°, 135°, 225°, and 315° all have the same reference angle at 45°
These angles all have thesame distance from theX-axis, so their Referenceangles are all equal
If we ever encounterthese angles, we can use Special triangles
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HOMEWORK:
Assignment 2.1