19.4 Angles and Arc Lengths 621 (1, 0); the line drawn from the origin to (0, 1) makes an angle of π 2 radians with the positive horizontal axis. v u π 2 π 2 v u 5π 2 v u –3π 4 Figure 19.22 Radians are unitless. When we write sin(1.1) and want to think of 1.1 as an angle, then it is an angle in radians. In other words, sin(1.1) = sin(1.1 radians). Converting Degrees to Radians and Radians to Degrees You may be accustomed to measuring angles in degrees, but radians are much more convenient for calculus. We need a way to go back and forth between the two measures. The fact that one full revolution is 360 ◦ ,or2πradians, allows us to do this. Equivalently, half a revolution is 180 ◦ ,orπradians. 180 ◦ = π radians π radians = 180 degrees 1 ◦ = π 180 radians 1 radian = 180 π degrees (≈ 57 ◦ ) EXERCISE 19.12 Convert from degrees to radians or radians to degrees. (a) 45 ◦ (b) −30 ◦ (c) 3π 2 radians (d) −2 radians Answers (a) π 4 (b) − π 6 (c) 270 ◦ (d) − 360 π ◦ ≈−114.59 ◦ Arc Length Arc lengths are usually described in terms of the angle that subtends the arc and the radius of the circle. We know that on a unit circle, an angle of x radians subtends an arc length of x units. What about on a circle of radius 3? The circumference of a circle of radius 3 is three times that of the unit circle, so the arc length subtended by an angle of x radians on the circle of radius 3 should be 3x. 622 CHAPTER 19 Trigonometry: Introducing Periodic Functions v u 3x 1.5x 1.51 angle of x radians x 3 3 Figure 19.23 An arc subtended by an angle of θ radians on a circle of radius r will have arc length = rθ. Notice that the formula for arc length is simple when the subtending angle is given in radian measure. The corresponding arc length formula for an angle of θ degrees is rθ ◦ · π 180 ◦ = r · θπ 180 . Trigonometric Functions of Angles We can think of the input of any trigonometric function as either a directed distance along the unit circle or as an angle, because an angle in standard position will determine a point on the unit circle. ◆ EXAMPLE 19.4 Suppose angle θ is in standard position and the point (a, b) lies on the terminal side of θ but does not lie on the unit circle. Determine the sine and cosine of θ. v u 1 (a, b) a R b R ( ( , Figure 19.24 SOLUTION Compute √ a 2 + b 2 to determine the distance from point (a, b) to the origin. Let’s denote this distance by R. The simplest approach is to just scale to a unit circle. If the point (a, b) lies on the terminal side of the angle, so does ( a R , b R ), and the latter point lies on the unit circle as well. 19.4 Angles and Arc Lengths 623 Therefore, cos θ = a R and sin θ = b R , where R = √ a 2 + b 2 . ◆ Tapping Circle Symmetry for Trigonometric Information Suppose we know that P = P(θ)=(0.8, 0.6), where θ ∈[0, 2π]. Then not only do we know the sine and cosine of θ, but we know the trigonometric functions of any angle coterminal with θ , that is, θ ±2πnfor n any integer. We actually know substantially more than this. The cosine and sine of θ are both positive, so θ must be an acute angle. Given the coordinates of point P , we also know the coordinates of points Q, R, and S (one in each of the remaining three quadrants), as shown in the Figure 19.25. v u P (θ) = (.8, .6) (–.8, .6) Q (–.8, –.6) R S (.8, –.6) θ v u P Q (π – θ) R (π + θ) S (2π – θ) θ θ θ θ Figure 19.25 Not only do we know the trigonometric functions of θ and any angle coterminal to θ ,but we also know the trigonometric functions of π − θ π + θ , 2π − θ, and any angle coterminal to any of these. 11 Use Figure 19.25 to verify the following. sin θ = 0.6 cos θ = 0.8 sin(π − θ)=0.6 cos(π − θ) =−0.8 sin(π + θ)=−0.6 cos(π + θ)=−0.8 sin(2π − θ)=−0.6 cos(2π − θ) =0.8 EXERCISE 19.13 Using only the symmetry of the unit circle and the information cos 1 ≈0.54 and sin 1 ≈0.84, approximate all solutions to the following equations. (a) cos x = 0.54 (b) cos x =−0.54 11 Actually we know more than this. By interchanging the u and v coordinates, we can obtain the trigonometric functions of π 2 − θ, π 2 + θ, 3π 2 − θ, and 3π 2 + θ. 624 CHAPTER 19 Trigonometry: Introducing Periodic Functions (c) sin x = 0.84 (d) sin x =−0.84 v u 1 1 P(1) Figure 19.26 Answers (a) x ≈ 1 + 2πn,where n is any integer or x ≈ (2π − 1) + 2πn, where n is any integer. (b) x ≈ (π − 1) + 2πn, where n is any integer or x ≈ (π + 1) + 2πn, where n is any integer. (c) x ≈ 1 + 2πn,where n is any integer or x ≈ (π − 1) + 2πn, where n is any integer. (d) x ≈ (2π − 1) + 2πn, where n is any integer or x ≈ (π + 1) + 2πn, where n is any integer. PROBLEMS FOR SECTION 19.4 1. (a) Convert the following to radians. (i) 60 ◦ (ii) 30 ◦ (iii) 45 ◦ (iv) −120 ◦ (b) Convert 2 radians to degrees. 2. Convert these angles to radian measure. (a) −60 ◦ (b) 45 ◦ (c) −270 ◦ (d) 40 ◦ (e) −120 ◦ 3. Convert these angles given in radians to degrees. (a) 3π 4 (b) −3π 4 (c) 5π 6 (d) 3π 2 (e) 5π 4 (f) −3.2 (g) 4 4. A second hand of a clock is 6 inches long. (a) How far does the pointer of the second hand travel in 20 seconds? (b) How far does the pointer of the second hand travel when the second hand travels through an angle of 70 ◦ ? 19.4 Angles and Arc Lengths 625 (c) In one hour the minute hand of the clock moves through an angle of 2π radians. In this amount of time, through what angle does the second hand travel? The hour hand? Give your answers in radians. 5. A bicycle gear with radius 4 inches is rotating with a frequency of 50 revolutions per minute. In 2 minutes what distance has been covered by a point on the corresponding chain? 6. A nautical mile is the distance along the surface of the earth subtended by an angle with vertex at the center of the earth and measuring 1 60 ◦ . (a) The radius of the earth is about 3960 miles. Use this to approximate a nautical mile. Give your answer in feet. (One mile is 5280 feet.) (b) The Random House Dictionary defines a nautical mile to be 6076 feet. Use this to get a more accurate estimate for the radius of the earth than that given in part (a). 7. The earth travels around the sun in an elliptical orbit, but the ellipse is very close to circular. The earth’s distance from the sun varies between 147 million kilometers at perihelion (when the earth is closest to the sun) and 153 million kilometers at aphelion (when the earth is farthest from the sun). Use the following simplifying assumptions to give a rough estimate of how far the earth travels along its orbit each day. Simplifying assumptions: Model the earth’s path around the sun as a circle with radius 150 million km. Assume that the earth completes a trip around the circle every 365 days. 8. A bicycle wheel is 26 inches in diameter. When the brakes are applied the bike wheel makes 2.2 revolutions before coming to a halt. How far has the bike traveled? (Assume the bike does not skid.) 9. A system of gears is set up as drawn. Q R Chain 5" 2" Consider the height of the point Q on the large gear. The height is measured as vertical position (in inches) with respect to the line through the center of the gears and is given as a function of time by h(t) = 5 cos(πt), where t is measured in seconds. 626 CHAPTER 19 Trigonometry: Introducing Periodic Functions (a) How long does it take for Q to make a complete revolution? (b) In 2 seconds how far does the chain travel? (c) How many times does the large gear rotate in 2 seconds? (d) How many times does the small gear rotate in 2 seconds? In 1 second? (e) Write an equation for the height of point R on the small gear as a function of time (in seconds). 10. The point (2, 3) lies on the terminal side of angle α when α is put in standard position. Label the coordinates of the points P(α); P(−α) and P(π +α) on the unit circle. Then find the following. (a) sin α (b) cos(α + 2π) (c) tan α (d) sin(−α) (e) cos(α + π) (f) sin(π − α) (g) tan(2π − α) 11. If sin θ = 5 13 and cos θ is negative, label the coordinates of the points P(θ), P(−θ) and P(π −θ) on the unit circle. Then find the following. (a) cos θ (b) tan θ (c) cos(−θ) (d) sin(θ + π) (e) tan(π − θ) 20 CHAPTER Trigonometry—Circles and Triangles 20.1 RIGHT-TRIANGLE TRIGONOMETRY: THE DEFINITIONS An Historical Interlude The word “trigonometry” has its origins in the words for “triangle” and “measurement,” but as yet we have not given much reason for referring to sin x , cos x , and tan x as “trigonometric” functions. In ancient civilizations the development of trigonometry was spurred on by problems in astronomy, surveying, construction, and navigation, in which people were concerned with measuring sides of triangles. Astronomers from numerous civilizations spearheaded the field. On papyruses from ancient Egypt and cuneiforms from Babylonia one can find evidence of the early development of trigonometry. The Greek astronomer Hipparchus, born in Iznik, Turkey, around 190 b.c., made great strides in the development of trigonometry, calculating what were essentially trigonometric tables by inscribing a triangle in a circle. 1 These were passed down to us in the Almagest, a treatise written by the Greek scientist Ptolemy around 100 a.d. This work was later translated and studied by Arabic and Hindu astronomers who, in the Middle Ages, furthered the field. Astronomers in India made major contributions to trigonometry in their work; Jai Singh’s observatories, such as Jantar Mantar in Jaipur, India, illustrate the remarkable degree of accuracy that had been obtained by the early 1700s. While the approach taken in the previous chapter defines trigonometric functions in terms of the coordinates of a point on the unit circle and exploits the periodic nature of such functions, the classical development of trigonometry focused on right triangles and the ratios of the lengths of their sides. This triangle perspective is important in navigation, surveying, 1 For more details, see Trigonometric Delights by Eli Maor, Princeton University Press, 1998, pp. 22–28. 627 628 CHAPTER 20 Trigonometry—Circles and Triangles optics, and astronomy. It is a valuable perspective whenever we analyze phenomena (such as force, velocity, or displacement) in terms of both magnitude and direction. The flexiblity of having two different viewpoints gives us added power in our use of trigonometric functions. Definitions Triangle trigonometry is based on the concept of similarity. Similar triangles have corre- sponding angles of equal measure and hence corresponding sides have proportional lengths; therefore the ratio of the lengths of pairs of corresponding sides of similar triangles is invari- ant. Consider a right triangle, 2 as drawn below. The side opposite the right angle is called the hypotenuse. The other two sides are referred to as the legs. The sum of the angles of any triangle is 180 ◦ ,orπradians; therefore in a right triangle the measure of one acute angle determines the measure of the other. 3 It follows that all right triangles with a particular acute angle are similar. Let θ be an acute angle in a right triangle. opp hyp adj θ Figure 20.1 We’ll denote the length of the side of the triangle opposite θ by opp, the length of the side of the triangle adjacent to θ by adj, and the length of the hypotenuse by hyp. Trigonometry uses the fact that ratios of pairs of sides of right triangles are functions of θ . Definitions of sin θ, cos θ, and tan θ are given below. They apply only to right triangles where θ is one of the acute angles. sin θ = opp hyp cos θ = adj hyp tan θ = opp adj For any acute angle θ, these definitions are in complete agreement with the circle definitions given in the previous chapter, as illustrated in Figure 20.2. 2 A right triangle is a triangle containing a right angle, an angle of π/2 radians or 90 degrees. 3 If θ is acute, then θ ∈ (0, π 2 );ifθis obtuse, then θ ∈ ( π 2 , π). 20.1 Right-Triangle Trigonometry: The Definitions 629 θ v u 1 1 adj opp (cos x, sin x) Figure 20.2 There are three more ratios that can be considered. Indeed, they have been not only considered but named. The functions cosecant, secant, and cotangent are the reciprocals of sin θ , cos θ , and tan θ , respectively. opp hyp adj θ sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp Figure 20.3 We can sketch the graphs of csc x, cos x, and cot x by beginning with the graphs of sin x, cos x, and tan x and sketching the reciprocal of each function. We know about the sign of the reciprocal: 1 n is positive if n is positive, negative if n is negative, undefined if n is zero. We know about the magnitude of the reciprocal: 1 n is > 1if|n|<1 the smaller |n|, the larger 1 n , < 1if|n|>1 the larger |n|, the smaller 1 n , = 1if|n|=1 (that is, if n =±1). 630 CHAPTER 20 Trigonometry—Circles and Triangles We arrive at the graphs drawn below. sec x cot x cos x xx –1 1 –π 2 –π 2 π 2 –3π 2 –3π 2 3π 2 3π 2 5π 2 sec x = cos x 1 cot x = sin x cos x csc x sin x x –1 1 –π –π π 2 ππ–2π –2π 2π2π csc x = sin x 1 Figure 20.4 Notice that csc x and sec x have period 2π , as do their reciprocals. The graphs of csc x and sec x hang off the graphs of sin x and cos x, attached at points where they take on a y-value of ±1; cot x is periodic with period π,asistanx. Youmay be wondering why trigonometric functions have names like sine and cosine, secant and cosecant, tangent and cotangent. The “co” prefix refers to complementary angles. By definition, two angles are complementary if their sum is π/2. Therefore, the acute angles of any right triangle are complementary. Suppose we denote these acute angles by θ and β. The leg of the triangle that is opposite θ is adjacent to β. Therefore, sin θ = cos β, tan θ = cot β, sec θ = csc β. In other words, the trigonometric function of θ is the “cofunction” of its complementary angle. hyp θ β θ and β are complementary angles Figure 20.5 EXERCISE 20.1 Show that sin x = cos( π 2 − x) for x ∈ [0, π 2 ). Conclude that sin x = cos(x − π 2 ); the graphs of sin x and cos x are horizontal translates. The drawing that follows indicates how the secant and tangent functions got their names. The circle is a unit circle, so the tangent segment AB has length tan θ and the secant segment OA has length sec θ . 4 4 For an explanation of how the remaining trigonometric functions got their names, see Eli Maor’s Trigonometric Delights, Princeton University Press, 1998, pp. 35–38. . and want to think of 1.1 as an angle, then it is an angle in radians. In other words, sin(1.1) = sin(1.1 radians). Converting Degrees to Radians and Radians to Degrees You may be accustomed to. trigonometric functions of θ and any angle coterminal to θ ,but we also know the trigonometric functions of π − θ π + θ , 2π − θ, and any angle coterminal to any of these. 11 Use Figure 19.25 to verify. pointer of the second hand travel when the second hand travels through an angle of 70 ◦ ? 19.4 Angles and Arc Lengths 625 (c) In one hour the minute hand of the clock moves through an angle of 2π