1. Section 2.1 Angles in Standard Position
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2. 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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3. II) Quadrants & X /Y axis
On the X-axis:
Right – Positive
Left - Negative
On the Y-axis:
Up – Positive
Bottom - Negative
Center: Origin (0,0)
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4. III) Unit Circles & Rotations
A “unit circle” is a circle rotated around the origin with a radius of 1 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)
Terminal Arm
Positive Direction
Negative Direction
Initial Arm
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5. 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 around the origin (counter-clockwise)
Ex: Draw the following angles in standard position: a) 62° b) 152 °
in standard position
in standard position
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6. Ex: Draw the following angles in standard position: a) 312° b) -77°
c) 450° d) 540°
450° in standard position
540° radians in standard position
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7. VI) Co-terminal Angles
Angles that have their terminal arms at the same position
Co-terminal angles have a difference of 360° or multiples of 360° (Full circles)
ie: 30°, 390°, 750°, °, °, -690°, ..etc
All these angles have the same
Position, “Co-terminal Angles”
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8. Ex: Given the following angles, provide two co-terminal angles and a formula for all the co-terminal angles
Add/subtract full circles
to get co-terminal angles
Add/subtract full circles
to get co-terminal angles
All co-terminal are sums/ differences of 360°
All co-terminal are sums/ differences of 360°
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9. VI) Reference Angles An angle created by the terminal arm and X-axis.
Reference angles must be in the same quadrant as the terminal arm
Ex: Given the following angles in standard position, find the
reference angle
in standard position
in standard position
Reference Angle
Reference Angle
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10. Ex: Given each of the following terminal arms, indicate which 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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11. Practice: Find the reference angle for each of the following angles in standard position:
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12. Practice: Find the reference angle for each of the following angles in standard position
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13. 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 the same distance from the X-axis, so their Reference angles are all equal
If we ever encounter these angles, we can use Special triangles
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14. Homework:
Assignment 2.1