Angles and Their Measure: Trigonometric Functions and Applications

Trigonometric Functions and Angles

Introduction to Trigonometric Functions

Trigonometric functions relate angles to ratios of side lengths in right triangles and to points on the unit circle. They are fundamental in measuring periodic phenomena and have broad applications in science and engineering.

• Trigonometric functions are used to relate the angles of a triangle to the lengths of its sides.

• They are important for understanding periodic phenomena, such as sound waves and circular motion.

Angles and Their Measure

Basic Definitions

An angle is formed by two rays (the sides of the angle) sharing a common endpoint (the vertex). Angles can be measured in degrees or radians.

• Ray: A part of a line that starts at a point and extends infinitely in one direction.

• Vertex: The common endpoint of the rays forming the angle.

• Initial side: The starting position of the ray.

• Terminal side: The position of the ray after rotation.

• Positive angle: Generated by counterclockwise rotation.

• Negative angle: Generated by clockwise rotation.

Standard Position of an Angle

• An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

• The terminal side determines the quadrant in which the angle lies.

Measuring Angles: Degrees and Radians

Angles can be measured in degrees (°) or radians (rad). One complete revolution is 360° or $2\pi$ radians.

• Degree: A unit of angle measure. 1 revolution = 360°.

• Radian: The measure of a central angle that intercepts an arc equal in length to the radius of the circle. 1 revolution = $2\pi$ radians.

Converting Between Degrees and Radians

• To convert degrees to radians, multiply by $\frac{\pi}{180}$.

• To convert radians to degrees, multiply by $\frac{180}{\pi}$.

Example: Convert 45° to radians:

$45^\circ \times \frac{\pi}{180} = \frac{\pi}{4}$ radians

Example: Convert $\frac{2\pi}{3}$ radians to degrees:

$\frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ$

Degrees, Minutes, and Seconds (DMS)

Angles can also be measured in degrees, minutes, and seconds, where 1 degree = 60 minutes and 1 minute = 60 seconds.

• To convert from decimal degrees to DMS, multiply the decimal part by 60 to get minutes, then multiply the decimal part of the minutes by 60 to get seconds.

• To convert from DMS to decimal degrees, divide minutes by 60 and seconds by 3600, then add to the degrees.

Example: Convert 42.5° to DMS:

42° 30' 0"

Angles on the Unit Circle

Drawing Angles in Standard Position

• Draw the initial side along the positive x-axis.

• Rotate counterclockwise for positive angles, clockwise for negative angles.

• The terminal side determines the quadrant.

Quadrants

• Quadrant I: 0° to 90° (0 to $\frac{\pi}{2}$ radians)

• Quadrant II: 90° to 180° ($\frac{\pi}{2}$ to $\pi$ radians)

• Quadrant III: 180° to 270° ($\pi$ to $\frac{3\pi}{2}$ radians)

• Quadrant IV: 270° to 360° ($\frac{3\pi}{2}$ to $2\pi$ radians)

Arc Length and Sector Area

Arc Length

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

$s = r\theta$

Example: Find the arc length of a circle of radius 5 m subtended by a central angle of 1.5 radians:

$s = 5 \times 1.5 = 7.5$ m

Area of a Sector

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

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

Example: Find the area of a sector of a circle of radius 3 feet formed by an angle of 40°:

First, convert 40° to radians: $40^\circ \times \frac{\pi}{180} = \frac{2\pi}{9}$ radians

$A = \frac{1}{2} \times 3^2 \times \frac{2\pi}{9} = \frac{9}{2} \times \frac{2\pi}{9} = \pi$ square feet

Linear and Angular Speed

Definitions

• Linear speed ($v$): The distance traveled per unit of time. $v = \frac{s}{t}$, where $s$ is arc length and $t$ is time.

• Angular speed ($\omega$): The rate of change of the angle per unit of time. $\omega = \frac{\theta}{t}$, where $\theta$ is in radians.

• Relationship: $v = r\omega$

Example: A child is spinning a rock at the end of a 2-foot rope at 110 revolutions per minute. Find the linear speed of the rock when it is released.

First, find the angular speed: $110 \text{ rev/min} \times 2\pi = 220\pi$ radians/min

Then, $v = r\omega = 2 \times 220\pi = 440\pi$ feet/min

Summary Table: Degree and Radian Conversions

Degrees	Radians
0°	0
30°	$\frac{\pi}{6}$
45°	$\frac{\pi}{4}$
60°	$\frac{\pi}{3}$
90°	$\frac{\pi}{2}$
120°	$\frac{2\pi}{3}$
135°	$\frac{3\pi}{4}$
150°	$\frac{5\pi}{6}$
180°	$\pi$
210°	$\frac{7\pi}{6}$
225°	$\frac{5\pi}{4}$
240°	$\frac{4\pi}{3}$
270°	$\frac{3\pi}{2}$
300°	$\frac{5\pi}{3}$
315°	$\frac{7\pi}{4}$
330°	$\frac{11\pi}{6}$
360°	$2\pi$