Angles and Radian Measure: Foundations of Trigonometry

Angles and Radian Measure

Definition of Angles

An angle is formed by two rays that share a common endpoint, called the vertex. One ray is the initial side, and the other is the terminal side.

• Initial Side: The starting position of the ray.

• Terminal Side: The position after rotation.

Angles in Standard Position

An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis.

• Quadrantal Angle: An angle whose terminal side lies on the x-axis or y-axis.

Positive and Negative Angles

• Positive Angles: Generated by counterclockwise rotation.

• Negative Angles: Generated by clockwise rotation.

Measuring Angles Using Degrees

Angles are measured by the amount of rotation from the initial side to the terminal side. One complete rotation of a circle is 360 degrees ($360^ ext{o}$).

• Acute Angle: Less than $90^ ext{o}$

• Right Angle: Exactly $90^ ext{o}$

• Obtuse Angle: More than $90^ ext{o}$ but less than $180^ ext{o}$

• Straight Angle: Exactly $180^ ext{o}$

Measuring Angles Using Radians

An angle whose vertex is at the center of a circle is called a central angle. The radian measure of a central angle is the length of the intercepted arc divided by the circle's radius.

Definition of a Radian

One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

Formula:

$\theta = \frac{s}{r}$

• $\theta$ = angle in radians

• $s$ = arc length

• $r$ = radius

Example: Computing Radian Measure

If a central angle $\theta$ in a circle of radius 12 feet intercepts an arc of length 42 feet:

$\theta = \frac{42}{12} = 3.5$ radians

Conversion Between Degrees and Radians

• Basic relationship: $\pi$ radians $= 180^\text{o}$

• To convert degrees to radians: $\text{radians} = \frac{\pi}{180^\text{o}} \times \text{degrees}$

• To convert radians to degrees: $\text{degrees} = \frac{180^\text{o}}{\pi} \times \text{radians}$

Examples

• Convert $60^\text{o}$ to radians: $\frac{\pi}{3}$ radians

• Convert $\frac{\pi}{4}$ radians to degrees: $45^\text{o}$

Drawing Angles in Standard Position

To draw an angle in standard position, start with the initial side on the positive x-axis and rotate to the terminal side. Positive angles rotate counterclockwise; negative angles rotate clockwise.

Degree and Radian Measures of Common Angles

Common angles in trigonometry are often expressed in both degrees and radians.

Degrees	Radians
0°	0
30°	$\frac{\pi}{6}$
45°	$\frac{\pi}{4}$
60°	$\frac{\pi}{3}$
90°	$\frac{\pi}{2}$
180°	$\pi$
270°	$\frac{3\pi}{2}$
360°	$2\pi$

Revolutions and Angle Measures

Terminal Side	Radian Measure	Degree Measure
1 revolution	$2\pi$	$360^\text{o}$
2 revolutions	$4\pi$	$720^\text{o}$
n revolutions	$2n\pi$	$360n^\text{o}$

Coterminal Angles

Coterminal angles are angles with the same initial and terminal sides but possibly different rotations.

• To find coterminal angles in degrees: Add or subtract multiples of $360^\text{o}$.

• To find coterminal angles in radians: Add or subtract multiples of $2\pi$.

Examples: Finding Coterminal Angles

• Find a positive angle less than $360^\text{o}$ coterminal with $400^\text{o}$: $400^\text{o} - 360^\text{o} = 40^\text{o}$

• Find a positive angle less than $2\pi$ coterminal with $\frac{13\pi}{5}$: $\frac{13\pi}{5} - 2\pi = \frac{3\pi}{5}$

The Length of a Circular Arc

The length $s$ of an arc intercepted by a central angle $\theta$ (in radians) in a circle of radius $r$ is:

$s = r\theta$

Example: Finding the Length of a Circular Arc

• Given $r = 6$ inches, $\theta = 45^\text{o} = \frac{\pi}{4}$ radians:

• $s = 6 \times \frac{\pi}{4} = 1.5\pi \approx 4.71$ inches

The Area of a Sector of a Circle

The area $A$ of a sector with radius $r$ and central angle $\theta$ (in radians):

$A = \frac{1}{2} r^2 \theta$

Example: Finding the Area of a Sector

• Given $r = 6$ feet, $\theta = 150^\text{o} = \frac{5\pi}{6}$ radians:

• $A = \frac{1}{2} (6)^2 \times \frac{5\pi}{6} = 18\pi \approx 47.12$ square feet

Definitions of Linear and Angular Speed

• Linear Speed ($v$): The rate at which a point moves along a circular path.

• Angular Speed ($\omega$): The rate at which the central angle changes, measured in radians per unit time.

Formulas:

• $v = \frac{s}{t}$ where $s$ is arc length, $t$ is time

• $\omega = \frac{\theta}{t}$ where $\theta$ is angle in radians, $t$ is time

• Relationship: $v = r\omega$

Example: Finding Linear Speed

• A record rotates at 45 revolutions per minute. At a point 1.5 inches from the center:

• Angular speed: $\omega = 45 \times 2\pi = 90\pi$ radians/minute

• Linear speed: $v = r\omega = 1.5 \times 90\pi = 135\pi$ inches/minute

Additional info: These notes provide foundational concepts for trigonometry, including angle measurement, radian and degree conversion, arc length, sector area, and speed calculations, all essential for further study in trigonometric functions and applications.