Radian Measure of Angles

Radians are essential in trigonometry for solving problems related to angles, arc lengths, and sector areas. A solid grasp of radians simplifies both mathematical calculations and practical applications.

Introduction to Radians

• Definition of Radians: A radian is the angle formed when an arc's length is equal to the circle's radius.
  • chatImportant

    Keep in Mind: A complete circle of 360° is equivalent to $2\pi$ radians.

• Comparison to Degrees: Conversions you should know:
  • $180^{\circ} = \pi$ radians
  • $90^{\circ} = \frac{\pi}{2}$ radians
  • $30^{\circ} = \frac{\pi}{6}$ radians
  • $60^{\circ} = \frac{\pi}{3}$ radians

Conversion Formula

• Degrees to Radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
• Radians to Degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Examples of Conversion

1. Convert 90° to Radians: $90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians
2. Convert 180° to Radians: $180 \times \frac{\pi}{180} = \pi$ radians
3. Convert $\frac{\pi}{2}$ Radians to Degrees: $\frac{\pi}{2} \times \frac{180}{\pi} = 90^{\circ}$
4. Convert $\pi$ Radians to Degrees: $\pi \times \frac{180}{\pi} = 180^{\circ}$

Unit Circle and Trigonometric Functions

• Unit Circle: A circle with a radius of 1 that links degrees, radians, and trigonometric values in Cartesian coordinates.
• Trigonometric Functions:
  • Sine ($\sin$): Y-coordinate
  • Cosine ($\cos$): X-coordinate
  • Tangent ($\tan$): The ratio $\frac{\sin(\theta)}{\cos(\theta)}$

Key Angles and Values

Angle (Degrees)	Angle (Radians)	$\sin$	$\cos$	$\tan$
0°	0	0	1	0
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$\frac{\pi}{2}$	1	0	undefined

Arc Length and Sector Area

• Arc Length Formula: $l = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians.
• Sector Area Formula: $A = \frac{1}{2}r^2\theta$

Example Problems

1. Given: $r = 5$, $\theta = \frac{\pi}{4}$.
   • Arc Length: $l = 5 \times \frac{\pi}{4} = \frac{5\pi}{4}$ units
   • Sector Area: $A = \frac{1}{2} \times 5^2 \times \frac{\pi}{4} = \frac{25\pi}{8}$ square units

Graphing Trigonometric Functions

• Key Functions:
  • $y = \sin x$
  • $y = \cos x$
  • $y = \tan x$
• Graph Characteristics:
  • Amplitude: Indicates the maximum height from the central axis.
  • Period: Describes the interval length over which the function repeats.
  • Phase Shift and Vertical Shift: Impact horizontal and vertical translations of the graph.

Example:

• For $y = \sin x$ with a period of $2\pi$: Main intercepts are at $(0,0)$, $(\pi,0)$, with peaks/troughs at $(\frac{\pi}{2},1)$, $(\frac{3\pi}{2},-1)$.

Visual Aids

Practice Problems

• Calculate the arc length for $r = 7$, $\theta = \frac{\pi}{3}$.

  • Solution: $l = r\theta = 7 \times \frac{\pi}{3} = \frac{7\pi}{3}$ units

• Determine the sector area for $r = 3$, $\theta = \frac{2\pi}{5}$.

  • Solution: $A = \frac{1}{2}r^2\theta = \frac{1}{2} \times 3^2 \times \frac{2\pi}{5} = \frac{9\pi}{5}$ square units