Directed Angles, Circular Geometry, and Measurement Systems

Directed Angles and Their Basic Notions

• Definition of a Directed Angle: Consider a ray $OA$. Rotate it about the vertex $O$ until it takes the position of ray $OB$. The angle obtained due to this rotation is called directed angle $AOB$. It is defined as the ordered pair of rays $(OA, OB)$ together with the rotation of the ray $OA$ to the position of the ray $OB$.
• Vertex and Arms:
  • Vertex: The point $O$ about which the ray rotates.
  • Initial Arm: In the ordered pair $(OA, OB)$, the ray $OA$ is the initial arm.
  • Terminal Arm: The ray $OB$ is the terminal arm.
• Direction of Rotation and Sign:
  • Positive Measure: If the rotation of the initial ray is anticlockwise, the measure of the directed angle is considered positive.
  • Negative Measure: If the rotation of the initial ray is clockwise, the measure of the directed angle is considered negative.
• Ordered Pair Dependency: The directed angle $(OA, OB) \neq (OB, OA)$ because the initial and terminal arms are different. Consequently, $\angle AOB \neq \angle BOA$ even if the amount of rotation is identical in magnitude.

Specific Types of Angles

• Zero Angle: If the ray $OA$ has zero rotation (it does not rotate), the initial arm itself is the terminal arm $OB$.
• One Rotation Angle: This is formed when the initial ray $OA$ performs one complete rotation and coincides with the terminal ray $OB$. The measure of one rotation angle is $360^\circ$.
• Straight Angle: Formed when the initial ray $OA$ and terminal ray $OB$ are in opposite directions after rotation. In this case, $AOB$ forms a straight line. A straight angle is half of one rotation angle ($180^\circ$).
• Right Angle: This is one-fourth of one rotation angle. It is also equal to half of a straight angle. One rotation angle contains four right angles ($90^\circ$ per right angle).

Angles in the Coordinate System

• Standard Position: In a rectangular coordinate system, a directed angle is in standard position if its vertex is at the origin $O$ and its initial ray lies along the positive X-axis. Examples include $\angle XOP$, $\angle XOQ$, and $\angle XOR$ if their initial sides are on the X-axis.
• Angle in a Quadrant: A directed angle in standard position is said to be in a particular quadrant if its terminal ray lies within that quadrant. For instance, if the terminal ray is in the first quadrant, the angle is in the first quadrant.
• Quadrantal Angles: A directed angle in standard position whose terminal ray lies specifically along the X-axis or Y-axis. Examples include measures like $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.
• Co-terminal Angles: Directed angles with different amounts of rotation that share the same initial ray and the same terminal ray.
  • Property: The difference between the measures of two co-terminal angles is always an integral multiple of $360^\circ$.
  • Example: Angles measuring $30^\circ, 390^\circ$, and $-330^\circ$ are co-terminal because they share the same initial and terminal arms. For instance: $390^\circ - (-330^\circ) = 720^\circ = 2 \times 360^\circ$.

Systems of Angle Measurement

• Sexagesimal System (Degree Measure):

  • The unit of measurement is the degree ($1^\circ$).
  • A single rotation angle is divided into $360$ equal parts; each part is $1^\circ$.
  • $1$ degree ($1^\circ$) = $\frac{1}{360}$th part of one complete rotation.
  • Subdivisions:
    • $1$ minute ($1'$) = $\frac{1}{60}$th part of one degree. Therefore, $1^\circ = 60'$.
    • $1$ second ($1''$) = $\frac{1}{60}$th part of one minute. Therefore, $1' = 60''$.
  • Standard Measures: One rotation = $360^\circ$; Straight angle = $180^\circ$; Right angle = $90^\circ$.

• Circular System (Radian Measure):

  • The unit of measurement is the radian ($1^c$).
  • Definition: One radian is the measure of a central angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
  • Theorem: The radian is a constant angle independent of the radius of the circle used.
  • Relation: $\pi^c = 180^\circ$.

Conversion Between Systems

• General Proportion: If an angle measures $r$ in radians and $\theta$ in degrees, the proportion to a straight angle is: $\frac{r}{\pi} = \frac{\theta}{180}$.
• Conversion Formulas:
  • To convert degree to radian: Multiply by $\frac{\pi}{180}$. Formula: $r^c = \theta^\circ \times \frac{\pi}{180}$.
  • To convert radian to degree: Multiply by $\frac{180}{\pi}$. Formula: $\theta^\circ = r^c \times \frac{180}{\pi}$.
• Specific Values:
  • $1^c = (\frac{180}{\pi})^\circ \approx 57.3248^\circ$.
  • Converting fractional degrees: $57.3248^\circ = 57^\circ 19' 29''$ approximately.
  • $15^\circ = \frac{\pi}{12}^c$
  • $30^\circ = \frac{\pi}{6}^c$
  • $45^\circ = \frac{\pi}{4}^c$
  • $60^\circ = \frac{\pi}{3}^c$
  • $90^\circ = \frac{\pi}{2}^c$
  • $120^\circ = \frac{2\pi}{3}^c$
  • $180^\circ = \pi^c$
  • $270^\circ = \frac{3\pi}{2}^c$
  • $360^\circ = 2\pi^c$

Clock Mathematics and Hand Rotation

• Minute Hand:
  • Completes one rotation ($1R$) in $60$ minutes.
  • $60\text{ min} = 360^\circ$.
  • In $1$ minute, the minute hand rotates through $6^\circ$.
• Hour Hand:
  • Completes one rotation ($1R$) in $12$ hours.
  • $12\text{ Hrs} = 360^\circ$.
  • In $1$ hour, the hour hand rotates through $30^\circ$.
  • In $1$ minute, the hour hand rotates through $\frac{30^\circ}{60} = 0.5^\circ$.
• Angle Between Hands Application:
  • Quarter past five (5:15): The minute hand is at $3$ ($90^\circ$ from 12). The hour hand has moved past $5$. At $5$, the angle is $5 \times 30^\circ = 150^\circ$. In $15$ minutes, the hour hand moves $15 \times 0.5^\circ = 7.5^\circ$. Position of hour hand = $150^\circ + 7.5^\circ = 157.5^\circ$. Difference = $157.5^\circ - 90^\circ = 67.5^\circ$.
  • Quarter to twelve (11:45): Minute hand at $9$ ($270^\circ$). Hour hand is near $12$. It will take $15$ minutes to reach $12$, which equals $7.5^\circ$. The angle from the $12$ mark to the hour hand is $7.5^\circ$. The angle between the hands is $90^\circ - 7.5^\circ = 82.5^\circ$.

Arc Length and Area of a Sector

• Arc Length ($s$):
  • The arc length $s$ is proportional to the central angle $\theta$ measured in radians.
  • Formula: $s = r\theta$.
• Area of a Sector ($A$):
  • The area $A$ of a sector is proportional to its central angle $\theta$ in radians.
  • Formula: $A = \frac{1}{2}r^2\theta$.
• Derivation Ratio:
  • For Arc Length: $\frac{\theta}{2\pi} = \frac{s}{2\pi r}$.
  • For Area: $\frac{\theta}{2\pi} = \frac{A}{\pi r^2}$.

Regular Polygons

• Interior and Exterior Angles:
  • For a regular polygon with $n$ sides, the sum of exterior angles is $360^\circ$.
  • Exterior angle = $(\frac{360}{n})^\circ$.
  • Interior angle = $180^\circ -$ Exterior angle.
• Example (Decagon): If each interior angle is $\frac{4\pi}{5}^c$ ($144^\circ$), the exterior angle is $180^\circ - 144^\circ = 36^\circ$. The number of sides $n = \frac{360}{36} = 10$.

Solved Problem Scenarios

• Triangle Angles in Arithmetic Progression (A.P.): If angles are $a-d, a, a+d$, their sum is $3a = 180^\circ$, so $a = 60^\circ$. If the smallest is $40^\circ$, then $60-d = 40$, so $d = 20$. Angles are $40^\circ, 60^\circ, 80^\circ$. Convert to radians: $\frac{2\pi}{9}^c, \frac{\pi}{3}^c, \frac{4\pi}{9}^c$.
• Right-Angled Triangle Differences: If acute angles $x$ and $y$ have a difference of $\frac{7\pi}{30}^c$ ($42^\circ$), then $x - y = 42$ and $x + y = 90$. Solving gives $2x = 132$, so $x = 66^\circ$ and $y = 24^\circ$. Total angles: $24^\circ, 66^\circ, 90^\circ$.
• Quadrilateral Ratios: If one angle is $\frac{2\pi}{9}^c$ ($40^\circ$), the sum of the other three is $360 - 40 = 320^\circ$. Given ratio $3:5:8$, angles are $3k, 5k, 8k$. $16k = 320$, so $k = 20$. Angles are $60^\circ, 100^\circ, 160^\circ$.
• Perimeter Conditions for Sectors: If the perimeter of a sector is half the circumference, then $2r + s = \pi r$. Since $s = r\theta$, then $2r + r\theta = \pi r$. Dividing by $r$, $2 + \theta = \pi \rightarrow \theta = (\pi - 2)^c$.