Objectives After completing this section, you should be able to:


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1 Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding of the relationship between arc length and the subtended angle. Find the angular velocity and linear speed of objects in rotational motion. Trigonometry The word trigonometry derives from the Greek: "trigono" for triangle (three angles), and "metron" for measure. Trigonometry is the branch of mathematics that deals with the relations of sides and angles of triangles, and with the relations among special functions associated with any angle. Trigonometry was developed by ancient cultures as a tool to help them map the apparent motion of stars and planets through the sky, and to help predict celestial phenomena such as the phases of the moon, eclipses, and equinoxes. This spherical trigonometry the study of spherical triangles on the surface of the socalled "celestial sphere" (pictured at right) was the first type of trigonometry to be discovered. Because of the curvature involved, spherical trigonometry is much more complex than planar trigonometry (the study of angles and angular relationships in planar figures), but it preceded the latter due to its astronomical application. The Greek mathematician Hipparchus of Rhodes ( BC) is considered the founder of trigonometry because he produced the first tables of geometric "chords" to be referenced in recorded history. In geometry, a chord is the straight segment that joins any two points in a circumference and, more generically, on any curve. continued Page 1 of 12
2 continued The figure below shows a chord as the segment between points A and B on the circumference of a circle. The curved portions of the circumference with A and B as starting and ending points are called arcs of the circumference. The space or opening between the radii that join the center of the circle with points A and B is called an angle. Note: This image represents an animation that can only be seen in the course online. As shown in the animation in the course online, the given points A and B on the circumference and the center of the circle define two angles, one much larger than the other but both sharing the same chord. The first chord tables produced by Hipparchus were for astronomical use and consisted of twelve books which, unfortunately, were lost. Only references to his work are found in documents by other mathematicians and astronomers. He was the first geometer to introduce the division of the circle into 360 equal angle sectors. Several books of chords in spherical and planar trigonometry were written by other astronomers and mathematicians in the following centuries. Ptolomy ( AD) built very complete tables of chords at intervals of 1/2 of a degree. Using Hipparchus' technique of dividing the circle into 360 equal sectors, and using the chords of the circle to construct the 360 sides of the inscribed regular polygon, he derived an approximation for the value of the number pi: Ptolomy recorded his ideas on geocentric planetary motion and trigonometric methods in his most important work titled Almagest, a treatise of thirteen books that dominated scientific knowledge for about fifteen centuries. Angles The geometric definition of an angle involves the use of rays, so we start by defining a ray: Rays are sometimes referred to as half lines. A ray is a straight line extending from a point P. continued Page 2 of 12
3 continued A geometric angle is the opening or space between two rays that share the same starting point, P. This point P is called the vertex of the angle, and the rays are the sides of the angle. A rotation angle is the opening or space defined when a ray rotates around a starting point P from an initial position to a final position. The position of the ray before rotation is the initial side of the angle, and the final position of the ray is its terminal side. Angles that are defined with the initial side coincident with the positive xaxis on the xy coordinate plane and the starting point P at the origin of the coordinate system are said to be in standard position. Note: This image represents an animation that can only be seen in the course online. Positive and Negative Angles Angles that are generated by rotating a ray counterclockwise are positive angles. Conversely, angles obtained by rotating the ray clockwise are negative angles, as shown in the figures below. The same convention applies when rotating a segment of a line around a fixed end from an initial position to a final (terminal) position. A complete rotation of such a segment, which is obtained when the terminal position coincides with the initial position of the segment, describes a full circle. Page 3 of 12
4 Angle Measurement Angles are traditionally measured in degrees, following the division of a circle into 360 equal sectors as done by Hipparchus. Thus, a full rotation describes an angle of 360 degrees (expressed as: 360 o ). When the two rays that define an angle are separated by exactly 90 degrees we say that they determine a right angle. By using this division into 360 equal sectors, we can find the geometric representation of various given angles. We can also do the reverse procedure: find the angle measure in degrees from a geometric representation of an angle. This is shown in the following examples. Example A: Draw the standard angle of each indicated measure using the 15degree grid shown. A) 60 B) 45 C) 180 D) 90 Since each division in the grid corresponds to 15 degrees, we start on the horizontal axis and rotate the terminal ray as many divisions as needed either counterclockwise or clockwise according to the sign of the angle. A) B) C) D) Page 4 of 12
5 Example B: Find the degree measure of each angle depicted below. A) B) A) We count the number of 15degree divisions that this positive (counterclockwise) angle encompasses: 14 sectors of 15 degrees = 210 degrees B) We count the number of 15degree divisions that this negative (clockwise) angle encompasses: 8 sectors of 15 degrees = 120 degrees Coterminal Angles Some angles, like the ones generated by the rotating rods shown in the figure below may have the same initial and terminal sides, although they were originated by different patterns. Such angles are called coterminal angles. Clearly, by adding a multiple of a full rotation, coterminal angles are obtained. For example, the 0 angle and the 360 angle are coterminal. Note: This image represents an animation that can only be seen in the course online. Page 5 of 12
6 Example C: Are the following standard angles coterminal angles? A) 105 degrees and 495 degrees. B) 150 degrees and 210 degrees A) No, these two angles are not coterminal angles since they do not share the same terminal position: B) Yes, these are coterminal angles since they share the same terminal position. Page 6 of 12
7 Special Angles There are some special angles which we will use extensively. These are the angles that mark the division of each quadrant of the xy plane into two and into three equal sectors. They are shown in the following figure. Note: This image represents an animation that can only be seen in the course online. Naming Angles When assigning names to generic angles, Greek letters are commonly used to recognize the work of ancient Greek mathematicians and geometers. Some of the Greek letters most commonly used to define angles are: Radians There is another common unit for measuring angles (different from the degree) called the radian. It is particularly convenient to use in advanced mathematics and in science because it greatly simplifies the derivation of trigonometric expressions. "Radian" is generally defined as the angle subtended by (opposite to) the arc of length equal to the circumference's radius (see the image below). A radian is equivalent to approximately The arc described by a segment (or "rod") rotating one full revolution (360 ) has a length of 2π ; therefore, the angle subtended corresponds to 2π radians. Similarly, a rod rotating half a revolution (180 ) defines an angle of π radians. continued Page 7 of 12
8 continued The table below shows the angle in both radians and degrees for fractions of a revolution that correspond to the special angles studied above. Note: This table represents an interactive practice exercise in the course online. The radian has become such a standard unit of angle measure that the word "radian" is not included after the numerical value when angles are expressed in this unit. Conversely, the "degree" unit must be specified after the numerical value for an angle to distinguish it from the radian, by using the degree symbol. For example: Using proportions and the equivalence between 180 and π radians, we can find very simple conversion formulas from radians to degrees and from degrees to radians, for any angle : continued Page 8 of 12
9 continued The mathematical procedure is summarized in the following image: Example D: Find the degree/radian conversion for the following angles: A) 105 to radians. B) 2 π radians to degrees. 5 π π 7π A) Multiply by 105 = radians B) Multiply by 180 π 2 π 180 o = π Note that the convention for positive and negative angles in standard position discussed earlier applies to radian measures as with degrees. The figure on the right shows the rectangular coordinate system with the quadrant boundaries labeled with 0, π /2, π and 3 π /2 radians. The positive angle in standard position subtended between the points 0 and P on the circumference and the origin of the coordinate system corresponds to π /6, equivalent to a + 30 angle. The negative angle pictured in orange subtended between the points 0 and R on the circumference and the origin of the coordinate system is π /6 (equivalent to an angle of 30 ). Page 9 of 12
10 Arc Length Considering a segment (or rod) that rotates around one of its ends to generate a rotation angle, we can find a mathematical expression that relates the subtended angle given in units of radians with the length of the arc described by the free end of the segment. Note: This image represents an animation that can only be seen in the course online. We can use simple proportionality to relate the length of the full circumference of radius r described by a rod in a full revolution about one of its ends and the subtended angle, 2 r, with the length of the arc (S) generated when the same rod rotates an angle : π The length of the arc S that subtends a central angle of radians in a circle of radius r is: S = θr That is, the length of the arc is the product of the radian measure of the angle times the radius of the circle. Example E: Find the length of the arc subtended by an angle of measure 108 on a circumference of radius 15 in. π 3π First convert the angle to radians: 108 = Then, multiply this radian measure times the radius of the circumference: 3π 15 in = 9π in in 5 Page 10 of 12
11 Angular Velocity In the same way that we define linear velocity for an object that covers a certain linear distance in a given time as: distance / time we can define angular velocity of an object describing circular motion as: central angle subtended / time An animation in the course online (an image of which is shown below) can help us understand angular velocity with the example of a fish swimming around a bowl. Note: This image represents an animation that can only be seen in the course online. The animation shows that we set up a coordinate system with the origin at the center of the circumference described by the fish's motion. We measure then the time it takes the fish to complete a full revolution around the origin of the coordinate system. The angular velocity is defined as: central angle subtended / time = 360 /8 sec = 45 / sec Angular velocities can be given in different units, depending on the units used to measure the angle and the time. Some common units are: degrees per hour, radians per second, and revolutions per minute. Page 11 of 12
12 Example F: Find the angular velocity in units of radians per second of a bicycle wheel that rotates eighty full revolutions in one minute. First convert the angle and the time to the requested units. The angle that the wheel rotates in radians is 80 2π = 160π. Then, express the time of the rotation in seconds 1 minute = 60 seconds. Now divide to obtain the angular velocity: 160 π 8 π = radians per 60 3 second. Apart from the angular velocity of a fish's rotational motion around a bowl, we can study the actual linear velocity (speed) of the fish in the water. The linear velocity associated with the rotational motion of the fish in our example on the previous page can be calculated by considering the length of the arc of circumference covered by the fish during rotation divided by the time. In this particular case, if we consider that the fish is swimming around a circle of radius r, then: We can conclude that when the angular velocity is known (with the angle measured in radians), the linear velocity is easily found by the following formula: linear velocity = angular velocity radius Example G: A trainer is trotting his horse around him. The horse trots in a circle whose radius is given by the length of the rope. The horse performs 3 full rotations in 20 seconds. If the length of the rope is 14 feet, find the linear speed (in feet per second) at which the horse is trotting. First, convert the angle described to radians 3 full rotations in radians is: 3 2π = 6π. Find the angular velocity by writing the quotient between the angle in radians and the time in seconds: 6 π 3 π = radians per second Now calculate the linear velocity by multiplying the angular speed times the radius of the rotational movement: 3 π π = feet per second End of Lesson Page 12 of 12
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