Podcast
Questions and Answers
What is the measure of a complete angle after one rotation?
What is the measure of a complete angle after one rotation?
- 90°
- 270°
- 360° (correct)
- 180°
A right angle measures 180°.
A right angle measures 180°.
False (B)
What is the term for directed angles with the same initial and terminal rays?
What is the term for directed angles with the same initial and terminal rays?
Co-terminal angles
The angle in standard position has its initial ray along the positive ______ axis.
The angle in standard position has its initial ray along the positive ______ axis.
Match the types of angles with their corresponding measures:
Match the types of angles with their corresponding measures:
Which of the following correctly describes directed angles?
Which of the following correctly describes directed angles?
The measure of the angle AOB is the same regardless of whether it is measured clockwise or anticlockwise.
The measure of the angle AOB is the same regardless of whether it is measured clockwise or anticlockwise.
What is the measure of angle ABC if it is drawn with a measure of 40°?
What is the measure of angle ABC if it is drawn with a measure of 40°?
In a directed angle, the ray _____ is called the initial arm.
In a directed angle, the ray _____ is called the initial arm.
Match the following angle characteristics:
Match the following angle characteristics:
What is the total number of degrees in one complete rotation?
What is the total number of degrees in one complete rotation?
Co-terminal angles can have the same measure.
Co-terminal angles can have the same measure.
What is the term used for an angle whose terminal ray lies along the x-axis or y-axis?
What is the term used for an angle whose terminal ray lies along the x-axis or y-axis?
The unit of measurement of angles in the sexagesimal system is called a ______.
The unit of measurement of angles in the sexagesimal system is called a ______.
Match the following terms with their correct descriptions:
Match the following terms with their correct descriptions:
How many degrees are there in one radian?
How many degrees are there in one radian?
One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
What is the formula to convert degrees into radians?
What is the formula to convert degrees into radians?
An angle measuring _____ radians corresponds to a straight angle.
An angle measuring _____ radians corresponds to a straight angle.
Which of the following radian measures corresponds to 45 degrees?
Which of the following radian measures corresponds to 45 degrees?
Match the following degree measures with their corresponding radian measures:
Match the following degree measures with their corresponding radian measures:
The conversion factor from radians to degrees is 180°/π.
The conversion factor from radians to degrees is 180°/π.
To convert 1 degree into minutes, you multiply the fractional part by _____ minutes.
To convert 1 degree into minutes, you multiply the fractional part by _____ minutes.
What is the formula to convert degrees into radians?
What is the formula to convert degrees into radians?
1° is equivalent to $57.3248$ radians.
1° is equivalent to $57.3248$ radians.
What is the radian measure of a 90° angle?
What is the radian measure of a 90° angle?
To convert radians into degrees, multiply the radian measure by ______.
To convert radians into degrees, multiply the radian measure by ______.
Which of the following is the correct conversion of 70° to radians?
Which of the following is the correct conversion of 70° to radians?
Match the degree measures with their corresponding radian measures:
Match the degree measures with their corresponding radian measures:
The term 'minute' in time measurement is the same as the 'minute' in angular measurement.
The term 'minute' in time measurement is the same as the 'minute' in angular measurement.
1R in a minute hand of a clock equals ______ degrees.
1R in a minute hand of a clock equals ______ degrees.
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Study Notes
Angle Basics
- An angle is formed by two rays (OA and OB) with a common endpoint (O).
- Directed angles designate orientation based on rotation: anticlockwise (positive) or clockwise (negative).
Types of Angles
- Zero Angle: No rotation, or $m(\angle AOB) = 0°$.
- Complete Angle: One full rotation, $m(\angle AOB) = 360°$.
- Straight Angle: Rays in opposite directions, $m(\angle AOB) = 180°$.
- Right Angle: $m(\angle AOB) = 90°$, one-fourth of a complete angle.
Angle Measurements
- Measured in degrees (sexagesimal system) or radians (circular system).
- Degrees: One complete rotation is 360°. Co-terminal angles differ by multiples of 360°.
- Radians: Defined such that an arc length equal to the radius subtends a central angle of 1 radian.
Radian Relations
- $1 \text{ radian} = \frac{\pi}{180°}$.
- Proportions for converting degrees to radians and vice versa:
- Degrees to Radians: Multiply by $\frac{\pi}{180}$.
- Radians to Degrees: Multiply by $\frac{180}{\pi}$.
Angles in Different Quadrants
- Quadrantal angles lie on the x-axis or y-axis (multiples of 90°).
- A directed angle exists in a quadrant if its terminal ray does.
Key Conversions
- Using $\pi \approx 3.14$ to approximate angles in degrees.
- Example: $1° \approx 57.3248°$ in decimal fractions.
Clock Angles
- The rotation of clock hands is analogous to angular measurements:
- 1 full rotation = 360° (both minute and hour hands).
- 1 minute hand rotation = 6°; 1 hour hand rotation = 30°.
Conversion Examples
- 70° to radians: $\frac{70}{180}\pi = \frac{7\pi}{18}$.
- -120° to radians: $\frac{-120}{180}\pi = \frac{-2\pi}{3}$.
- $\frac{1}{4}°$ to radians: $\frac{1}{4} \times \frac{\pi}{180} = \frac{\pi}{720}$.
Arc and Sector Areas
- The length of an arc and the area of a sector relate to the central angle and the circle's radius, applying the angular measurements directly.
Application Activities
- Practice drawing directed angles and converting between degrees and radians for reinforcement of concepts.
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