Sine and Cosine functions

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The Unit Circle: Sine and Cosine Functions Figure 1 The tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr) ChAPTeR OUTl Ine 7.1 Angles 7.2 Right Triangle Trigonometry 7.3 Unit Circle 7.4 The Other Trigonometric Functions Introduction Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. Similarly, the progression from day to night occurs as a result of Earth’s rotation, and the pattern of the seasons repeats in response to Earth’s revolution around the sun. Outside of nature, many stocks that mirror a company’s prots a fi re influenced by changes in the economic business cycle. In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape ree fl ctive of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions. 575576 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns l eARnIng Obje CTIveS In this section, you will: • Draw angles in standard position. • Convert between degrees and radians. • Find coterminal angles. • Find the length of a circular arc. • Use linear and angular speed to describe motion on a circular path. 7.1 AngleS A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the die ff rence between success and failure in many undertakings. In this section, we will examine properties of angles. drawing Angles in Standard Position Properly defining an angle first requires that we define a ray. A ray consists of one point on a line and all points extending in one direction from that point. e Th first point is called the endpoint of the ray. We can refer to a specic fi → ray by stating its endpoint and any other point on it. The ray in Figure 1 can be named as ray EF, or in symbol form EF. Ray EF F Endpoint E Figure 1 An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the → → two rays are the sides of the angle. The angle in Figure 2 is formed from ED and EF. Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form ∠DEF. Angle DEF D F E Vertex Figure 2 Greek letters are often used as variables for the measure of an angle. Table 1 is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 3. θ φ or ϕ α β γ theta phi alpha beta gamma Table 1 θ Figure 3 Angle theta, shown as ∠θSECTION 7.1 aN gl 577 Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the die ff rent sides, we indicate the rotation with a small arc and arrow close to the vertex as in Figure 4. Terminal side Vertex Initial side Figure 4 As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to 1 _ the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is o f a circular 360 rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90°. To formalize our work, we will begin by drawing angles on an x-y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See Figure 5. Standard Position y Terminal side x Initial side Figure 5 If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle. Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do 90° 1 _ _ that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that = . 360° 4 So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive 360° _ x-axis. To draw a 360° angle, we calculate that = 1. So the terminal side will be 1 complete rotation around the 360° circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap. See Figure 6. Drawing a 90° angle Drawing a 360° angle yy 360° Terminal Initial side 90° side xx Initial side Terminal side Figure 6 es578 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns Since we define an angle in standard position by its terminal side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of 0°, 90°, 180°, 270° or 360°. See Figure 7. II I II I II I II I 0° 90° 180° 270° III IV III IV III IV III IV Figure 7 Quadrantal angles have a terminal side that lies along an axis. examples are shown. quadrantal angles An angle is a quadrantal angle if its terminal side lies on an axis, including 0°, 90°, 180°, 270°, or 360°. How To… Given an angle measure in degrees, draw the angle in standard position. 1. Express the angle measure as a fraction of 360°. 2. Reduce the fraction to simplest form. 3. Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles. Example 1 Drawing an Angle in Standard Position Measured in Degrees a. Sketch an angle of 30° in standard position. b. Sketch an angle of −135° in standard position. Solution a. Divide the angle measure by 360°. 30° 1 ____ __ = 360° 12 To rewrite the fraction in a more familiar fraction, we can recognize that 1 1 1 __ __ __ =   12 3 4 One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30° as in Figure 8. y 30° x Figure 8SECTION 7.1 aN gl 579 b. Divide the angle measure by 360°. −135° 3 _____ __ = − 360° 8 In this case, we can recognize that 3 3 1 __ __ __ −   = −     8 2 4 Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure 9. y x –135° Figure 9 Try It 1 Show an angle of 240° on a circle in standard position. Converting between degrees and Radians Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle. The circumference of a circle is C = 2πr. If we divide both sides of this equation by r, we create the ratio of the circumference to the radius, which is always 2π regardless of the length of the radius. So the circumference of any circle is 2π ≈ 6.28 times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure 10. 3 2 4 1 Fractional piece 5 6 Figure 10 es580 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals 2π times the radius, a full circular rotation is 2π radians. So 2π radians = 360° 360° ____ π radians = = 180° 2 180° ____ 1 radian = ≈ 57.3° π See Figure 11. Note that when an angle is described without a specic u fi nit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out. r r t Figure 11 The angle t sweeps out a measure of one radian. note that the length of the intercepted arc is the same as the length of the radius of the circle. Relating Arc Lengths to Radius An arc length s is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius. This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length s to the radius r. See Figure 12. s = rθ s _ θ =   r r _ If s = r, then θ =   = 1 radian. r 2 radians 1 radian 2r 3 radians r s r A full revolution 0, 2π 1 radian (s = r) 6 radians 4 4 radians 5 radians (c) (a)(b) Figure 12 (a) In an angle of 1 radian, the arc length s equals the radius r. (b) An angle of 2 radians has an arc length s = 2r. (c) A full revolution is 2π or about 6.28 radians. To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is C = 2πr, where r is the radius. The smaller circle then has circumference 2 π(2) = 4π and the larger has circumference 2π(3) = 6π. Now we draw a 45°angle on the two circles, as in Figure 13.SECTION 7.1 aN gl 581 π 45° = − radians 4 45° r = 3 r = 2 Figure 13 A 45° angle contains one-eighth of the circumference of a circle, regardless of the radius. Notice what happens if we find the ratio of the arc length divided by the radius of the circle. 1 __ π 2 1 _ __ Smaller circle: =   π 2 4 3 __ π 4 1 _ __ Larger circle: =   π 3 4 1 _ Since both ratios are π, the angle measures of both circles are the same, even though the arc length and radius die ff r. 4 radians One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360°) equals 2π radians. A half revolution (180°) is equivalent to π radians. The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if s is the length of an arc of a circle, and r is the radius of the circle, then the central angle s _ containing that arc measures radians. In a circle of radius 1, the radian measure corresponds to the length of r the arc. Q & A… A measure of 1 radian looks to be about 60°. Is that correct? 360° _ Yes. It is approximately 57.3°. Because 2π radians equals 360°, 1 radian equals ≈ 57.3°. 2π Using Radians Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 11, suppose the radius was 2 inches and the distance along the arc was also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure. Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, C = 2πr, and for the unit circle C = 2π. These two die ff rent ways to rotate around a circle give us a way to convert from degrees to radians. 1 rotation = 360° = 2π radians 1 __ rotation = 180° = π radians 2 1 π __ __ rotation = 90° = radi ans 4 2 es582 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns Identifying Special Angles Measured in Radians In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure 14. Memorizing these angles will be very useful as we study the properties associated with angles. 90° π 2 2π π 120° 60° 3 90° 3 3π π 135° 45° 60° 120° 4 4 135° 45° 150° 5π π 30° 6 6 150° 30° 180° 0° π 180º 0° 2π 210° 330° 210° 330° 7π 11π 6 6 225° 315° 225° 315° 5π 7π 300° 240° 240° 300° 4 4 5π 270º 4π 3 270° 3π 3 2 Figure 14 Commonly encountered angles measured in degrees Figure 15 Commonly encountered angles measured in radians Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 14, which are shown in Figure 15. Be sure you can verify each of these measures. Example 2 Finding a Radian Measure Find the radian measure of one-third of a full rotation. Solution For any circle, the arc length along such a rotation would be one-third of the circumference. We know that 1 rotation = 2πr So, 1 __ s =    (2 πr) 3 2πr ___ = 3 e r Th ad ian measure would be the arc length divided by the radius. 2πr _ 3 _ radian measure = r 2πr ___ = 3r 2π ___ = 3 Try It 2 Find the radian measure of three-fourths of a full rotation. Converting Between Radians and Degrees Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion where θ is the measure of the angle in degrees and θ is the measure of the angle in radians. R θ θ R _ _ = π 180 This proportion shows that the measure of angle θ in degrees divided by 180 equals the measure of angle θ in radians divided by π. Or, phrased another way, degrees is to 180 as radians is to π. Degrees Radians _ _ = π 180SECTION 7.1 aN gl 583 converting between radians and degrees To convert between degrees and radians, use the proportion θ θ R _ _ = π 180 Example 3 Converting Radians to Degrees Convert each radian measure to degrees. π __ a. b. 3 6 Solution Because we are given radians and we want degrees, we should set up a proportion and solve it. a. We use the proportion, substituting the given information. θ θ R _ _ = π 180 π __   θ 6 _ _ =   π 180 180 _ θ = 6 θ = 30° b. We use the proportion, substituting the given information. θ θ R _ _ = π 180 θ 3 _ _ = π 180 3(180) _ θ = π θ ≈ 172° Try It 3 3π _ Convert − radians to degrees. 4 Example 4 Converting Degrees to Radians Convert 15 degrees to radians. Solution In this example, we start with degrees and want radians, so we again set up a proportion and solve it, but we substitute the given information into a die ff rent part of the proportion. θ θ R _ _ = π 180 15 θ R _ _ = π 180 15π _ = θ R 180 π _ = θ R 12 π 1 _ _ Analysis Another way to think about this problem is by remembering that 30° =    . Becau se 15° =    (3 0°), we can find that 6 2 π π 1 _ _ _ is .   2 6 12 es584 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns Try It 4 Convert 126° to radians. Finding Coterminal Angles Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0° to 360°, or 0 to 2π. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution. It is possible for more than one angle to have the same terminal side. Look at Figure 16. The angle of 140° is a positive angle, measured counterclockwise. The angle of −220° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than 360° or less than 0° is coterminal with an angle between 0° and 360°, and it is often more convenient to find the coterminal angle within the range of 0° to 360° than to work with an angle that is outside that range. y 140° x –220° Figure 16 An angle of 140° and an angle of –220° are coterminal angles. Any angle has infinitely many coterminal angles because each time we add 360° to that angle—or subtract 360° from it— the resulting value has a terminal side in the same location. For example, 100° and 460° are coterminal for this reason, as is −260°. An angle’s reference angle is the measure of the smallest, positive, acute angle t formed by the terminal side of the angle t and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 17 for examples of reference angles for angles in die ff rent quadrants. Quadrant I Quadrant II Quadrant III Quadrant IV y y y y II I II I II I II I t' t t t t x x x x t' t' III IV III IV III IV III IV t' = tt' = π − t = 180° − t π t' = 2π − t = 360° − t t' = t − = t − 180° Figure 17 coterminal and reference angles Coterminal angles are two angles in standard position that have the same terminal side. An angle’s reference angle is the size of the smallest acute angle, t′, formed by the terminal side of the angle t and the horizontal axis.SECTION 7.1 aN gl 585 How To… Given an angle greater than 360°, find a coterminal angle between 0° and 360°. 1. Subtract 360° from the given angle. 2. If the result is still greater than 360°, subtract 360° again till the result is between 0° and 360°. 3. e r Th esulting angle is coterminal with the original angle. Example 5 Finding an Angle Coterminal with an Angle of Measur e Greater Than 360° Find the least positive angle θ that is coterminal with an angle measuring 800°, where 0° ≤ θ 360°. Solution An angle with measure 800° is coterminal with an angle with y measure 800 − 360 = 440°, but 440° is still greater than 360°, so we subtract 360° again to find another coterminal angle: 440 − 360 = 80°. 80° e a Th ngle θ = 80° is coterminal with 800°. To put it another way, 800° equals 800° 80° plus two full rotations, as shown in Figure 18. x Figure 18 Try It 5 Find an angle α that is coterminal with an angle measuring 870°, where 0° ≤ α 360°. How To… Given an angle with measure less than 0°, find a coterminal angle having a measure between 0° and 360°. 1. Add 360° to the given angle. 2. If the result is still less than 0°, add 360° again until the result is between 0° and 360°. 3. e r Th esulting angle is coterminal with the original angle. Example 6 Finding an Angle Coterminal with an Angle Measuring Less Than 0° Show the angle with measure −45° on a circle and find a positive coterminal angle α such that 0° ≤ α 360°. Solution Since 45° is half of 90°, we can start at the positive horizontal axis y and measure clockwise half of a 90° angle. Because we can find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°: −45° + 360° = 315° x –45° We can then show the angle on a circle, as in Figure 19. 315° Figure 19 Try It 6 Find an angle β that is coterminal with an angle measuring −300° such that 0° ≤ β 360°. es586 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns Finding Coterminal Angles Measured in Radians We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations. How To… Given an angle greater than 2π, find a coterminal angle between 0 and 2 π. 1. Subtract 2π from the given angle. 2. If the result is still greater than 2π, subtract 2π again until the result is between 0 and 2π. 3. e r Th esulting angle is coterminal with the original angle. Example 7 Finding Coterminal Angles Using Radians 19π _ Find an angle β that is coterminal with , where 0 ≤ β 2π. 4 Solution When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of 2 π radians: 19π 19π 8π ___ ___ ___ − 2π = − 4 4 4 11π ___ = 4 11π _ e a Th n gle is coterminal, but not less than 2π, so we subtract another rotation: 4 11π 11π 8π ___ ___ ___ − 2π = − 4 4 4 3π ___ = 4 3π 19π _ _ e a Th n gle i s coterminal with , a s shown in Figure 20. 4 4 y 19π 4 x 3π 4 Figure 20 Try It 7 17π _ Find an angle of measure θ that is coterminal with an angle of measure −   where 0 ≤ θ 2π. 6 determining the l ength of an Arc Recall that the radian measure θ of an angle was defined as the ratio of the arc length s of a circular arc to the radius s _ r of the circle, θ = . From this relationship, we can find arc length along a circle, given an angle. rSECTION 7.1 aN gl 587 arc length on a circle In a circle of radius r, the length of an arc s subtended by an angle with s measure θ in radians, shown in Figure 21, is s = rθ θ r Figure 21 How To… Given a circle of radius r, calculate the length s of the arc subtended by a given angle of measure θ. 1. If necessary, convert θ to radians. 2. Multiply the radius r by the radian measure of θ : s = r θ. Example 8 Finding the Length of an Arc Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun. a. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day? b. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day. Solution a. Let’s begin by finding the circumference of Mercury’s orbit. C = 2πr = 2π(36 million miles) ≈ 226 million miles Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled: (0.0114)226 million miles = 2.58 million miles b. Now, we convert to radians: arclength _ radian = radius 2.58 million miles __ = 36 million miles = 0.0717 Try It 8 Find the arc length along a circle of radius 10 units subtended by an angle of 215°. Finding the Area of a Sector of a Circle In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius r θ 2 _ can be found using the formula A = πr . If the two radii form an angle of θ, measured in radians, then is the ratio 2π of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the θ _ area of the circle. Thus, the area of a sector is the fraction multiplied by the entire area. (Always remember that this 2π formula only applies if θ is in radians.) θ __ 2 Area of sector =  πr   2π 2 θπr ____ = 2π 1 __ 2 = θr 2 es588 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns area of a sector The area of a sector of a circle with radius r subtended by an angle θ, 1 2 measured in radians, is A = θ r r 2 1 __ 2 A =    θ r θ 2 See Figure 22. Figure 22 The area of the sector equals half the square of the radius times the central angle measured in radians. How To… Given a circle of radius r, find the area of a sector defined by a given angle θ. 1. If necessary, convert θ to radians. 1 _ 2 2. Multiply half the radian measure of θ by the square of the radius r : A =   θ r . 2 Example 9 Finding the Area of a Sector An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 23. What is the area of the sector of grass the sprinkler waters? 20  30° Solution First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert: Figure 23 The sprinkler sprays 20 ft within an arc of 30°. π ___ 30 degrees = 30 · 180 π __ = radians 6 Th e area of the sector is then π 1 __ __ 2 Area = (20)   2 6 ≈ 104.72 2 So the area is about 104.72 ft . Try It 9 In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places. Use l inear and Angular Speed to describe motion on a Circular Path In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10π inches, every second. So the linear speed of the point is 10π in./s. The equation for linear speed is as follows where v is linear speed, s is displacement, and t is time. s _ v =    tSECTION 7.1 aN gl 589 Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes 360 degrees __ a full rotation every 4 seconds, we can calculate its angular speed as = 90 degrees per second. Angular 4 seconds speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where ω (read as omega) is angular speed, θ is the angle traversed, and t is time. θ __ ω =    t Combining the definition of angular speed with the arc length equation, s = rθ, we can find a relationship between angular and linear speeds. The angular speed equation can be solved for θ, giving θ = ωt. Substituting this into the arc length equation gives: s = rθ = rωt Substituting this into the linear speed equation gives: s _ v=    t rωt ___ = t = rω angular and linear speed As a point moves along a circle of radius r, its angular speed, ω, is the angular rotation θ per unit time, t. θ __ ω =    t The linear speed, v, of the point can be found as the distance traveled, arc length s, per unit time, t. s _ v =   t When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation v = rω This equation states that the angular speed in radians, ω, representing the amount of rotation occurring in a unit of time, can be multiplied by the radius r to calculate the total arc length traveled in a unit of time, which is the definition of linear speed. How To… Given the amount of angle rotation and the time elapsed, calculate the angular speed. 1. If necessary, convert the angle measure to radians. θ _ 2. Divide the angle in radians by the number of time units elapsed: ω =   . t 3. e r Th esulting speed will be in radians per time unit. Example 10 Finding Angular Speed A water wheel, shown in Figure 24, completes 1 rotation every 5 seconds. Find the angular speed in radians per second. Solution The wheel completes 1 rotation, or passes through an angle of 2 π 2π _ radians in 5 seconds, so the angular speed would be ω = ≈ 1.257 radians 5 per second. Figure 24 es590 CHAPTER 7 t he uN it c ircl e: s i Ne a Nd c osi Ne f u Nctio Ns Try It 10 An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second. How To… Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed. 1. Convert the total rotation to radians if necessary. θ _ 2. Divide the total rotation in radians by the elapsed time to find the angular speed: apply ω =   t. 3. Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply v = rω. Example 11 Finding a Linear Speed A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road. Solution Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road. We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion: rotations 2π radians radians _______ ________ ______ 180 · = 360π minute rotation minute Using the formula from above along with the radius of the wheels, we can find the linear speed: radians ______ v = (14 inches) 360π   minute inches ______ = 5040π minute Remember that radians are a unitless measure, so it is not necessary to include them. Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour. inches 1 feet 1 mile 60 minutes ______ _______ _______ _________ 5040π · · · ≈ 14.99 miles per hour (mph) minute 12 inches 5280 feet 1 hour Try It 11 A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour. Access these online resources for additional instruction and practice with angles, arc length, and areas of sectors. • Angles in Standard Position (http://openstaxcollege.org/l/standardpos) • Angle of Rotation (http://openstaxcollege.org/l/angleofrotation) • Coterminal Angles (http://openstaxcollege.org/l/coterminal) • determining Coterminal Angles (http://openstaxcollege.org/l/detcoterm) • Positive and negative Coterminal Angles (http://openstaxcollege.org/l/posnegcoterm) • Radian measure (http://openstaxcollege.org/l/radianmeas) • Coterminal Angles in Radians (http://openstaxcollege.org/l/cotermrad) • Arc l ength and Area of a Sector (http://openstaxcollege.org/l/arclength)SECTION 7.1 s ectio N e xercises 591 7.1 SeCTIOn exe RCISeS veRbAl 1. Draw an angle in standard position. Label the vertex, 2. Explain why there are an infinite number of angles initial side, and terminal side. that are coterminal to a certain angle. 3. State what a positive or negative angle signifies, and 4. How does radian measure of an angle compare to the explain how to draw each. degree measure? Include an explanation of 1 radian in your paragraph. 5. Explain the differences between linear speed and angular speed when describing motion along a circular path. For the following exercises, draw an angle in standard position with the given measure. 2π ___ 6. 30° 7. 300° 8. −80° 9. 135° 10. −150° 11. 3 7π 5π π π ___ ___ __ __ 16. 415° 17. − 120° 12. 13. 14. 15. − 4 6 2 10 22π π 4π ___ __ ___ 18. − 315° 19. 20. −  21. −   3 6 3 For the following exercises, refer to Figure 25. Round to For the following exercises, refer to Figure 26. Round to two decimal places. two decimal places. r = 3 in 2π 5 140° r = 4.5 cm 23. Figure 25 Figure 26 24. 22. Find the arc length. 24. Find the arc length. 23. Find the area of the sector. 25. Find the area of the sector. Algeb RAIC For the following exercises, convert angles in radians to degrees. 5π 3π π π ___ ___ __ __ 28. − radians 26. radians 27. radians  29. radians 4 4 9 3 7π 5π 11π ___ ___ ___ 30. −   radians 31. −   radians 32. radians 3 12 6 For the following exercises, convert angles in degrees to radians. 33. 90° 34. 100° 35. −540° 36. − 120° 37. 180° 38. −315° 39. 150° For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places. 40. Find the length of the arc of a circle of radius 41. Find the length of the arc of a circle of radius π π _ _ 12 inches subtended by a central angle of radians. 5.02 miles subtended by the central angle of . 4 3 42. Find the length of the arc of a circle of diameter 43. Find the length of the arc of a circle of radius 5π _ 10 centimeters subtended by the central angle of 50°. 14 meters subtended by the central angle of . 6 44. Find the length of the arc of a circle of radius 45. Find the length of the arc of a circle of diameter 5 inches subtended by the central angle of 220°. 12 meters subtended by the central angle is 63°.592 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns For the following exercises, use the given information to find the area of the sector. Round to four decimal places. 46. A sector of a circle has a central angle of 45° and a 47. A sector of a circle has a central angle of 30° and a radius 6 cm. radius of 20 cm. 48. A sector of a circle with diameter 10 feet and an 49. A sector of a circle with radius of 0.7 inches and an π _ angle of radians. angle of π radians. 2 For the following exercises, find the angle between 0° and 360° that is coterminal to the given angle. 50. − 40° 51. − 110° 52. 700° 53. 1400° For the following exercises, find the angle between 0 and 2 π in radians that is coterminal to the given angle. π 10π 13π 44π __ ___ ___ ___ 54. −   55. 56. 57. 9 3 6 9 ReAl-W ORld A PPl ICATIOnS 58. A truck with 32-inch diameter wheels is traveling at 59. A bicycle with 24-inch diameter wheels is traveling at 60 mi/h. Find the angular speed of the wheels in 15 mi/h. Find the angular speed of the wheels in rad/ rad/min. How many revolutions per minute do the min. How many revolutions per minute do the wheels wheels make? make? 60. A wheel of radius 8 inches is rotating 15°/s. What is 61. A wheel of radius 14 inches is rotating 0.5 rad/s. What the linear speed v, the angular speed in RPM, and the is the linear speed v, the angular speed in RPM, and the angular speed in rad/s? angular speed in deg/s? 62. A CD has diameter of 120 millimeters. When playing 63. When being burned in a writable CD-R drive, the audio, the angular speed varies to keep the linear angular speed of a CD varies to keep the linear speed speed constant where the disc is being read. When constant where the disc is being written. When writing reading along the outer edge of the disc, the angular along the outer edge of the disc, the angular speed of speed is about 200 RPM (revolutions per minute). one drive is about 4,800 RPM (revolutions per minute). Find the linear speed. Find the linear speed if the CD has diameter of 120 millimeters. 64. A person is standing on the equator of Earth (radius 65. Find the distance along an arc on the surface of Earth that subtends a central angle of 5 minutes 3960 miles). What are his linear and angular speeds? 1 _ 1 minute = degree . The radius of Earth is 3,960 mi .   60 66. Find the distance along an arc on the surface of 67. Consider a clock with an hour hand and minute hand. Earth that subtends a central angle of 7 minutes What is the measure of the angle the minute hand 1 _ traces in 20 minutes? 1 minute = degree . The radius of Earth is   60 3,960 miles. exTen SIOnS 68. Two cities have the same longitude. The latitude of city 69. A city is located at 40 degrees north latitude. Assume A is 9.00 degrees north and the latitude of city B is 30.00 the radius of the earth is 3960 miles and the earth degree north. Assume the radius of the earth is 3960 rotates once every 24 hours. Find the linear speed of a miles. Find the distance between the two cities. person who resides in this city. 70. A city is located at 75 degrees north latitude. Assume 71. Find the linear speed of the moon if the average the radius of the earth is 3960 miles and the earth distance between the earth and moon is 239,000 miles, rotates once every 24 hours. Find the linear speed of a assuming the orbit of the moon is circular and requires person who resides in this city. about 28 days. Express answer in miles per hour. 72. A bicycle has wheels 28 inches in diameter. A 73. A car travels 3 miles. Its tires make 2640 revolutions. tachometer determines that the wheels are rotating at What is the radius of a tire in inches? 180 RPM (revolutions per minute). Find the speed the bicycle is travelling down the road. 74. A wheel on a tractor has a 24-inch diameter. How many revolutions does the wheel make if the tractor travels 4 miles?SECTION 7.2 r ight t ri a Ngle t rig o Nome ytr 593 l eARnIng Obje CTIveS In this section, you will: • Use right triangles to evaluate trigonometric functions. π π π _ _ _ • Find function values for 30° , 45° , and 60° .       6 4 3 • Use equal cofunctions of complementary angles. • Use the definitions of trigonometric functions of any angle. • Use right-triangle trigonometry to solve applied problems. 7.2 RIgh T TRIAngle T RIgOn OmeTRy Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. e Th measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains. Using Right Triangles to evaluate Trigonometric Functions Figure 1 shows a right triangle with a vertical side of length y and a horizontal side has length x. Notice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle. (x, y) 1 y t x Figure 1 We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. e Th adjacent side is the side closest to the angle, x. (Adjacent means “next to.”) The opposite side is the side across from the angle, y. The hypotenuse is the side of the triangle opposite the right angle, 1. These sides are labeled in Figure 2. Hypotenuse Opposite t Adjacent Figure 2 The sides of a right triangle in relation to angle t. Given a right triangle with an acute angle of t, the first three trigonometric functions are listed. opposite adjacent opposite __ __ _ Sine sin t = Cosine cos t = Tangent tan t = hypotenuse hypotenuse adjacent A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.” For the triangle shown in Figure 1, we have the following. y y x _ _ _ sin t =   cos t =   tan t =   x 1 1594 CHAPTER 7 t he uN it c irc le: s i Ne a Nd c osi Ne f u Ncti o Ns How To… Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle. 1. Find the sine as the ratio of the opposite side to the hypotenuse. 2. Find the cosine as the ratio of the adjacent side to the hypotenuse. 3. Find the tangent as the ratio of the opposite side to the adjacent side. Example 1 Evaluating a Trigonometric Function of a Right Triangle Given the triangle shown in Figure 3, find the value of cos α. 17 8 α 15 Figure 3 Solution The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17. adjacent _ cos(α) = hypotenuse 15 __ = 17 Try It 1 Given the triangle shown in Figure 4, find the value of sin t. 25 7 t 24 Figure 4 Reciprocal Functions In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle. hypotenuse hypotenuse adjacent __ __ _ Secant sec t = Cosecant csc t = Cotangent cot t = opposite opposite adjacent Take another look at these definitions. These functions are the reciprocals of the first three functions. 1 1 _ _ sin t =   csc t =   csc t sin t 1 1 _ _ cos t =   sec t =   sec t cos t 1 1 _ _ tan t =   cot t =   cot t tan t When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in Figure 5. e s Th ide opposite one acute angle is the side adjacent to the other acute angle, and vice versa.