APPENDIX A: Trigonometry Basics

Transcription

1 APPENDIX A: Trigonometr Basics Trigonometr Basics A Degree Measure A Right-Triangle Trigonometr A4 Unit-Circle Trigonometr A Radian Measure A0 Angle Measure Conversions A4 Sine and Cosine Functions A5 A. Concept Inventor A9 A. Activities A9 Answers to Appendi A A5 Copright Houghton Mifflin Compan. All rights reserved.

2 A APPENDIX A Trigonometr Basics This appendi gives supplementar material on degree measure and right-triangle trigonometr. It can be used in connection with Section.. Degree Measure One complete revolution of a circle is divided into 360 equal parts called degrees. Talking about small parts of a rotation is awkward, so we call partial rotations angles. The measure of an angle is described b the amount of rotation in the turn. For eample, a 90-degree angle is one-fourth of a complete counterclockwise revolution (see Figure A.). A small circle as a superscript on the angle measure is used as a smbol to indicate degrees (90 ). 90 FIGURE A. A 90 angle is 4 of a complete revolution. Degree Measurement 360 degrees 360 complete revolution degree of a complete revolution 360 The selection of 360 as the number of divisions is rooted in histor and ma reflect the fact that the Earth completes one revolution about the Sun in approimatel 365 das. Because 365 does not have a large number of divisors (onl 5 and 73), the inventors of the sstem probabl picked 360 with its multitude of divisors (, 3, 4, 5, 6,, 9, 0,, 5,, 0, 4, 30, 36, 40, 45, 60, 7, 90, 0, and 0) as being close to the number of das of the ear and eas to use in computing. The fortunate choice of 360 gives us a large number of even-degree angles that are simple fractions of a full revolution.* *It is possible to divide one revolution into 400 equal parts. Incidentall, some do. These parts are called gradians or grads, and this angle measure is available on man calculators. Of course, ou might prefer to be especiall patriotic and divide one revolution into 776 equal parts. In this case, ou would be inventing a new angle measure. Copright Houghton Mifflin Compan. All rights reserved.

3 Trigonometr Basics A3 EXAMPLE Measuring Angles in Degrees a. Epress of a complete rotation in degrees. b. Epress 3 of a complete rotation in degrees. Solution Keeping in mind that one complete rotation is 360, we have a. (360 ) 45 (See Figure A.a.) b. 3(360 ) 0 (See Figure A.b.) 45 0 (a) of a revolution (b) 3 of a revolution is a 45 angle. is a 0 angle. FIGURE A. In the stud of trigonometr, we consider revolutions around a particular circle. Specificall, we consider the unit circle that is, the circle with radius, centered at the origin. We sa that an angle is in standard position when the verte of the angle is at the origin and one of its sides is drawn along the positive -ais. The side that is drawn on the positive -ais is called the initial side. The other side of the angle is called the terminal side. Refer to Figure A.3. Figure A.4 shows a 90 angle, a 60 angle, and a 50 angle. Terminal side θ Radius = Initial side FIGURE A.3 FIGURE A.4 Because a 90 angle is one-quarter of a rotation (one quarter of the wa around the full circle), a 0 angle would be one-half of a rotation, a 70 angle would be Copright Houghton Mifflin Compan. All rights reserved.

4 A4 APPENDIX A FIGURE A three-quarters of a rotation, and a 360 angle is one full rotation. Note that in Figure A.4, the 60 angle is drawn between a quarter rotation and a half rotation, and the 50 angle is drawn between a half rotation and a three-quarter rotation. You should also note that all of the rotations are drawn in the counterclockwise direction. We consider counterclockwise rotation to define a positive angle and clockwise rotation to define a negative angle. For eample, Figure A.5 illustrates a 90 angle and a 00 angle. When angle measures are larger than 360, the describe more than one rotation around the circle. For eample, an 0 angle describes two full rotations plus onequarter of a rotation in the counterclockwise direction ( ). A 505 angle describes one full rotation plus one-quarter of a rotation plus an etra 55 angle in the clockwise direction ( ). These two angles are shown in Figure A FIGURE A.6 Before defining trigonometric functions for angles on a unit circle, we take a brief look at the trigonometric functions defined in terms of a right triangle. Hpotenuse θ Leg adjacent to θ FIGURE A.7 Leg opposite θ Right-Triangle Trigonometr From the earl use of trigonometr (meaning triangle measurement ) through present-da applications, right triangles have provided a method for solving problems that involve indirect measurements. A right triangle is a triangle with one 90 angle. In applications, we generall label one of the remaining angles of the right triangle as angle. (See Figure A.7.) We define the sine of the angle as the ratio of the length of the leg opposite the angle to the length of the hpotenuse (the side opposite the right angle). The cosine of the angle is the ratio of the length of the adjacent leg to the length of the hpotenuse. The tangent of the angle is the ratio of the length of the leg opposite the angle to the length of the adjacent leg. We abbreviate cosine b cos, sine b sin, and tangent b tan. Copright Houghton Mifflin Compan. All rights reserved.

5 Trigonometr Basics A5 Right-Triangle Trigonometric Definitions For a right triangle with one of the non-90 angles having measure, the sine, cosine, and tangent of are defined in terms of the sides of the triangle. sin cos tan length of the leg opposite the angle length of the hpotenuse length of the leg adjacent to the angle length of the hpotenuse length of the leg opposite the angle length of the leg adjacent to the angle These three functions are illustrated in Figure A.. θ Hpotenuse opposite leg sin θ = hpotenuse Opposite leg θ Hpotenuse Adjacent leg adjacent leg cos θ = hpotenuse θ Adjacent leg opposite leg tan θ = adjacent leg Opposite leg FIGURE A. Three other functions that are commonl studied in trigonometr are the secant, cosecant, and cotangent functions. The are defined as follows: length of the hpotenuse secant : sec cos length of the adjacent leg length of the hpotenuse cosecant : csc sin length of the opposite leg length of the adjacent leg cotangent : cot tan length of the opposite leg We stud primaril sin, cos, and tan because the other three trigonometric functions are defined in terms of these first three. If we are given an right triangle for which we know the angle and know the length of one side of the triangle, we can use the trigonometric functions sin, cos, and tan to obtain the lengths of the other two sides of the triangle. For eample, if we have a right triangle with a 5 angle and hpotenuse of length inches, we can use sin 5 and cos 5 to find the lengths of the two legs. (See Figure A.9.) Copright Houghton Mifflin Compan. All rights reserved.

6 A6 APPENDIX A inches b 5 a FIGURE A.9 A. To find the length of the leg adjacent to the 5 angle, we use the fact that cos 5 is the ratio of the length of the adjacent leg to the length of the hpotenuse. We will use a a to represent the length of the adjacent leg, so cos 5. Solving for a, we have a cos 5 (0.9063).6 inches That is, the leg adjacent to the 5 angle is approimatel. inches long. Similarl, the length of side b can be found as b sin 5 (0.46) inch Trigonometr is often used in applications where indirect measurement of the length of one or more legs of a triangle is necessar. Land surveing often relies on the use of an instrument called a setant to measure angles and on trigonometr to determine distances using right triangles. EXAMPLE Using Trig to Determine Measurement Castle Measurement A soldier in ancient times making use of triangle ratios is illustrated in Figure A.0. He is using a setant and sighting it in line with the top of the castle wall. The setant gives an angle of 4.6. The soldier then counts the bricks in the wall and estimates the distance d without having to enter the battle zone between his safe spot and the castle wall. h θ = 4.6 d FIGURE A.0 a. Estimate the height of the wall if each brick is inches tall. b. Find the distance to the castle. c. Find the distance from the soldier to the top of the castle wall. Copright Houghton Mifflin Compan. All rights reserved.

7 Trigonometr Basics A7 Solution a. There are laers of bricks in the wall, so the wall is ( bricks)( inches per brick) 4 inches 0. feet tall. b. Using the definition of the tangent of an angle, we know that 0. feet tan 4.6 d so we have 0. feet 0. feet d 77.4 feet tan The soldier is approimatel 77.4 feet awa from the castle. c. The Pthagorean Theorem ields h feet The soldier is approimatel 0 feet awa from the top of the castle wall. Two special right triangles that occur in applications using trigonometr are the right triangle and the right triangle. The values of sine, cosine, and tangent for one of these triangles are eplored in the net eample. The other triangle is left for an activit. EXAMPLE 3 s 60 s 30 h s 60 FIGURE A. Determining Ratios for a Special Triangle The right triangle is so named because in a right triangle with a 30 angle, the remaining angle must be 60. A reflection of the triangle across the leg between the 30 angle and the right angle forms an equilateral triangle (one whose three sides are of equal length). The line of reflection is the perpendicular bisector of the base. (See Figure A..) a. Use the Pthagorean Theorem to find the length h in terms of s. b. Use the sides h and s to find the sine, cosine, and tangent for the 30 angle in a right triangle. c. Use the sides h and s to find the sine, cosine, and tangent for the 60 angle. Solution a. The side opposite the 30 angle has length s that is equal to half of the length s of the hpotenuse. Appling the Pthagorean Theorem ields h s (s) Solving for h, we find the remaining leg length to be h (s) s 4s s 3s 3s Copright Houghton Mifflin Compan. All rights reserved.

8 A APPENDIX A 60 s s 30 3s FIGURE A. b. Referring to Figure A., we find the values of sine, cosine, and tangent for the 30 angle in the right triangle. sin 30 cos 30 tan 30 c. Similarl, the values of sine, cosine, and tangent for the 60 angle follow: sin 60 cos 60 s s 3s s s 3s 3s s s s s tan 60 3 s Unit-Circle Trigonometr θ FIGURE A.3 Right triangle associated with angle Right-triangle trigonometr is useful in man applications. However, as we previousl noted, the trigonometric functions are not restricted to use with angles that are less than 90. We can now define the trigonometric functions sine, cosine, and tangent for all angles b referring to our knowledge of right-triangle trigonometr. When the angle is between 0 and 90, we draw a right triangle in the unit circle such that the hpotenuse of the triangle is along the terminal side of the angle from the origin to the unit circle, and the other two legs of the right triangle are drawn along the -ais and perpendicular to the -ais, as shown in Figure A.3. If we let (, ) represent the point where the terminal edge of angle intersects the unit circle, then the leg drawn along the -ais has length and the leg drawn perpendicular to the -ais has length. The hpotenuse has length (the radius of the unit circle). We define the trigonometric functions sin, cos, and tan as we did for the right triangle, so that sin cos tan We define the trigonometric functions similarl for angles that are not between 0 and 90. First, draw the angle on the unit circle. Then draw a right triangle such that the hpotenuse of the triangle is along the terminal side of the angle from the origin to the unit circle, and the other two legs of the right triangle are drawn along Copright Houghton Mifflin Compan. All rights reserved.

9 Trigonometr Basics A9 the -ais (possibl in the negative direction) and perpendicular to the -ais (possibl in the negative direction) as shown in Figures A.4a and b. (, ) = (-a, b) b -a θ -d -c θ (, ) = (-c, -d) (a) (b) FIGURE A.4 Once again, call the point where the terminal edge of angle intersects the unit circle (, ), and let a represent the length of the leg drawn along the -ais and let b represent the length of the leg drawn perpendicular to the -ais. The hpotenuse has length (the radius of the unit circle). However, we must indicate whether the legs of the triangle are drawn in the negative and/or the negative direction. We indicate this b writing the negative of the length in the appropriate direction. For eample, the triangle in Figure A.4a has one leg along the negative portion of the -ais. This leg has length a in the negative direction, so a. The other leg is drawn up from the -ais; thus it has positive direction and has length h. We define the trigonometric functions sin, cos, and tan as we have done previousl, so that sin, cos, and tan. Thus, for the angle b b drawn in Figure A.4a, sin b, cos a, and tan a a. Similarl, for the d d angle drawn in Figure A.4b, sin d, cos c, and tan c c. In general, for an angle, we define sine, cosine, and tangent as follows: Unit-Circle Trigonometric Definitions Let (, ) be the point on the unit circle where the terminal side of the angle intersects the circle. Then the sine, cosine, and tangent of the angle are sin cos tan A. Because the unit circle is given b the equation, it is true that for an angle,cos sin. Copright Houghton Mifflin Compan. All rights reserved.

10 A0 APPENDIX A EXAMPLE 4 Determining Ratios for a Specific Angle -40 A 40 angle is shown in Figure A.5. a. Draw the right triangle associated with a 40 angle. Label the angle in the triangle between the terminal side and the -ais. b. Find the sine, cosine, and tangent of a 40 angle. Solution a. FIGURE A.5 60 FIGURE A.6 Right triangle associated with the 40 angle b. This right triangle is one of the special triangles mentioned in Eample 3. It is a right triangle. We know from Eample 3 that sin 60 and cos 60. Thus the leg on the -ais is positive and has length cos 60, so. The other leg is drawn down from the -ais, so it s measure is negative and 3 3 has length sin 60, so. Place these values on Figure A.6. (See Figure A.7.) Now we can read the trigonometric values from the figure: sin(40 ) 3-3 cos(40 ) tan(40 ) 3 FIGURE A.7 We have now defined trigonometric functions as the appl to right triangles and, in the much broader sense, as the appl to unit circles. However, our discussion of unit-circle trigonometr would not be complete if we did not consider another angle measure called radian measure. In fact, we cannot use the trig functions in calculus without considering radian measure of angles. Radian Measure Another unit of angle measurement is called a radian. Picture a circular pizza. We can describe a slice of pizza in terms of the angle of the wedge formed b the slice. For Copright Houghton Mifflin Compan. All rights reserved.

11 Trigonometr Basics A FIGURE A. 6 of a pizza is a 60 wedge. r FIGURE A.9 The radian measure of an angle is the ratio of the arc length to the radius. s r s instance, in angular terms we would describe one-eighth of a pizza as a 45 wedge. A one-sith slice would be described as a 60 wedge (see Figure A.). Another wa to think of a slice of pizza is in terms of the amount of crust around the edge. If there were something special in the crust, like a cheese filling, then we might want to focus on the length of the crust around the circular edge of the slice. In doing so, we would be talking about an arc of a circle that is, the slice s portion of the circumference. (A portion of the circle is called an arc, and the complete distance around the circle is its circumference.) If the radius r of the pizza in Figure A. were foot, then the circumference of the pizza would be r ( foot) feet 6.3 feet. Basic geometr tells us that the arc length is the same fractional part of the circumference of the circle as the angle is of one complete rotation. Thus the wedge would form an arc of 360 ( feet) 0. foot of the special cheese crust. The wedge would form an arc of 360 ( feet) foot. We define the radian measure of an angle to be the ratio of the length of the arc s cut out of the circumference of a circle b the angle to the radius r of the circle. (See Figure A.9.) Because the total arc length (that is, the circumference) of a circle is r times its radius, a full revolution has radian measure of r. (See Figure A.0.) Note that is a real number approimatel equal to 6.3. When we are using a unit circle, the radius is, so the radian measure is simpl the corresponding arc s s length: r s. It is important to recognize that the radian measure of an angle is a real number with no units attached. In the definition of radian measure, both s and r measure length and must have the same units, which causes to be a unitless quantit. However, we sometimes report this measure as radians. Radian Measurement radians complete counterclockwise rotation radian counterclockwise rotation (a little less than 6 revolution) = π radians = radian (a) (b) FIGURE A.0 Figure A. shows some common radian measures on a circle. Copright Houghton Mifflin Compan. All rights reserved.

12 A APPENDIX A 3π 4 π π 4 π 0, π 5π 4 3π 7π 4 FIGURE A. The angle corresponds to of a complete rotation, or () 4 units around the 3 circle. Thus its radian measure is 4. EXAMPLE 5 Understanding Radian Measure of Angles -5 a. Draw angles with radian measure 4 and 5 on a unit circle. b. Estimate the radian measure of the two angles shown in Figures A.a and b. (a) (b) FIGURE A. = 5π 4 (a) FIGURE A.3 = 5 (b) Solution -5 a. An angle of 4 4 ( ) ( ) is half of a clockwise rotation plus another of a clockwise rotation 3 (see Figure A.3a). An angle of 5 is between of a counterclockwise rotation and a complete rotation ( 6.3) (see Figure A.3b). Copright Houghton Mifflin Compan. All rights reserved.

13 Trigonometr Basics A3 b. The angle in Figure A.a appears to be approimatel of a rotation, or 0.5 radian. The angle in Figure A.b is 3: one complete clockwise rotation () plus a half of a clockwise rotation (). EXAMPLE 6 Measuring Angles in Radians a. Epress of a complete rotation in radians. b. Epress 4 of a complete rotation in radians. c. Epress 4 radians in terms of complete rotations. Solution Keeping in mind that complete rotation is radians, we have a. () 4 radians b. 4() radians c. Because radian of a complete revolution, 4 radians is 4 complete rotations. The odometer on an automobile converts turns of the drive shaft into distances so that ou can observe the number of miles ou have traveled. Although the net eample eplores this idea, it is not unique to automobiles. Whenever wheels rotate to cause movement, the same tpe of relationship between rotations of the wheels and distance traveled holds true. EXAMPLE 7 Using Radius to Determine Arc Length Rotating Wheels Consider a front-wheel-drive automobile with wheels that are 0 inches in diameter. a. How far will rotation of the wheels cause the automobile to travel? b. How man times will the wheels revolve when the automobile travels mile? Solution a. As an automobile travels, its drive shaft and front tires turn at the same rate; that is, revolution in one causes revolution in the other. If the wheels are 0 inches in diameter (radius 0 inches), then each rotation causes the automobile to move forward r ()(0 inches) 6. inches 5. feet. Each rotation of the wheels causes the automobile to travel about 6. inches or 5. feet. feet inches b. There are 50 mile foot 63,360 inches in a mile. So inches revolution revolutions 63, mile 0 inches mile The wheels must revolve approimatel 00.4 times in each mile traveled. Copright Houghton Mifflin Compan. All rights reserved.

14 A4 APPENDIX A Angle Measure Conversions Whenever ou have two measurement scales that can be applied to the same object, it is important to have a conversion technique. In this case, revolution is both 360 and radians. Thus degrees are converted to radians b multipling b 360 0, 0 and radians are converted to degrees b multipling b. Our conversion formula is Angle Conversion Formula radians 0 Up to this point, we have used the word radian to specif our choice of angle measure, not the units of the angle. You ma have noted that it is cumbersome to write the word radians each time angle measure is used. Therefore, we adopt the following convention. Angle Measure Convention All angles are understood to be measured in radians unless the degree smbol is used to specif degree measure of the angle. That is, an angle with measure 60 is quite different than one with measure 60. An angle of 60 radians is more than 9 full revolutions, whereas an angle of 60 is onl one-sith of a revolution. If ou mean an angle of 60 degrees, be certain that ou use the degree smbol with the angle. EXAMPLE A.3a, b Converting Angles a. 7 Epress 6 in degrees. b. Epress 35 in radians. Solution a b. 35 (35 ) 0 4 Copright Houghton Mifflin Compan. All rights reserved.

15 Trigonometr Basics A5 Sine and Cosine Functions We now define sine and cosine functions of angles in standard position on a unit circle. The bo also contains a ver important trig identit that follows from the fact that the equation of the unit circle at the origin of the - and -aes is. Trigonometric Values for a General Angle Consider an angle in standard position. The terminal side of the angle intercepts the unit circle at a point (, ). The sine and (, ) = (cos, sin ) cosine functions are defined as follows: f() sin is the function whose output is the -coordinate. g() cos is the function whose output is the FIGURE A.4 -coordinate. Identit: cos sin for an angle (, 0) Figure A.5 shows the four points where the unit circle intersects the aes, the corresponding angles in radians, and the values of the sine and cosine functions. = _ π, cos _ π = 0, sin _ π = (0, ) = π, cos π = -, sin π = 0 (-, 0) (, 0) = 0, cos 0 =, sin 0 = 0 (0, -) = 3π _, 3π _ cos = 0, sin 3π _ = - FIGURE A.5 An point on the unit circle has multiple corresponding angles with the same initial 5 and terminal sides. For eample, an angle of is complete revolution plus a quarter of a revolution, resulting in the same point on the unit circle as the 5 angle. Hence cos cos 0. Copright Houghton Mifflin Compan. All rights reserved.

16 A6 APPENDIX A The values of the sine and cosine functions are not obvious for points other than those shown in Figure A.5. You can use a calculator or computer to find other sine and cosine values, as illustrated in Eample. Be sure our technolog is set in radian mode. EXAMPLE 9 Instructions for using technolog with this eample are given in Section.. of the technolog supplement. Calculating Sine and Cosine Values 9 9 a. Find sin and cos. b. Interpret our answer to part a in terms of the coordinates of a point on the unit circle. c. Find two other angles whose sine and cosine are the same as those in part a. Solution 9 a. Using a calculator or computer ou should find that sin 0.36 and 9 cos π _ (-0.93, -0.36) FIGURE A.6 b. The cosine and sine values are the - and -coordinates of the point where the terminal side of the angle intersects the unit circle. This angle and the correspond- 9 ing point are shown in Figure A.6. Note that because the - and -coordinates of the point are negative, both the cosine and the sine of this angle are negative. c. We seek two other angles corresponding to the point where the terminal side of the angle shown in Figure A.6 intersects the unit circle. One such angle is 9 obtained b a clockwise rotation from the positive -ais. This angle is 7 and is epressed as a negative number to denote the direction. Thus the angle is, and sin sin and cos cos. It is also possible to reach the point under consideration b going around the 9 circle full rotation counterclockwise and then an additional radians. This angle is. Thus sin sin and cos cos. There are Copright Houghton Mifflin Compan. All rights reserved.

17 Trigonometr Basics A7 infinitel more positive and negative angles corresponding to the point indicated in Figure A.6, all with a sine value of approimatel 0.36 and a cosine value of approimatel So far we have referred to f() sin and g() cos as functions, but we have not verified that this is the case. To verif that f() sin is a function, we must ask, Can sin have more than one output value for a particular angle (the input)? Because the terminal side of the angle intersects the unit circle at a single point, there will be onl one output f() sin corresponding to each angle. Thus f() sin is indeed a function. Similar reasoning confirms that g() cos is also a function. Be careful not to confuse angle inputs of these functions. Even though angles 3 such as 6 and 6 have the same initial and terminal sides and therefore the same trigonometric function outputs, the two angles are not the same: 6 is of a 3 revolution, whereas 6 represents complete revolution plus additional revolution. For cclic functions, infinitel man distinct inputs correspond to the same output. In a graph of the sine function, the horizontal ais represents the angle measures and the vertical ais represents the -coordinates on the unit circle. For the cosine function, the horizontal ais represents the angle measure and the vertical ais represents the -coordinate on the unit circle. Refer to Figure A.5. If we plot the angles shown in that figure and the corresponding -coordinates, we obtain the graph shown in Figure A.7. We also add the point (, 0) corresponding to complete revolution. We add more points to this graph b calculating some intermediate values for the sine function. Table A. shows selected angles between 0 and (corresponding to siteenths of a revolution) and the associated -coordinates (to five decimal places) on a unit circle. Figure A. shows the points in Table A. added to the graph in Figure A.7. - f () π _ π FIGURE A.7 Points on the graph of f() sin 3π _ π TABLE A. sin sin Copright Houghton Mifflin Compan. All rights reserved.

18 A APPENDIX A f () f () f () = sin π _ π 3π _ π π _ π 3π _ π - - FIGURE A. FIGURE A.9 Because the -coordinates increase, decrease, and increase again and take on all real number values between and as we move around the circle, we epect the sine function graph to increase and decrease in a smooth, continuous manner, taking on all values between and. We therefore connect these points with a smooth curve to obtain the graph of the sine function shown in Figure A.9. Once the angle eceeds, we begin retracing the unit circle, and the sine function begins repeating itself. Figure A.30 shows a sine graph with more of the repetitions. This graph etends infinitel far in both directions. f () f () = sin -π -3π _ -π -_ π _ π π 3π _ π 5π _ 3π 7π _ 4π - FIGURE A.30 The sine function is periodic because it repeats itself ever input units. The period of the sine function is. The sine function is also cclic, because it varies continuousl, alternating between and. The portion of the sine function over one period that is, the part that keeps repeating itself is called a ccle of the function. Although we have viewed the sine and cosine functions as having angle measure as input, we are not restricted to this interpretation. In fact, we can consider the sine and cosine functions as having an real number as the input. Because most applications to which we will appl a sine model have input that is not an angle measure, we will no longer use to denote the input. Instead we use the notation sin and cos. Do not confuse and here with points on the unit circle. The variable is simpl the input, which can be interpreted as an angle measured in radians, and the variable is the output, which can be interpreted as the - or -coordinate on the unit circle, depending on whether we are considering a cosine function or a sine function. Copright Houghton Mifflin Compan. All rights reserved.

19 Trigonometr Basics A9 A. Concept Inventor A. Activities Degree measure of angles Initial and terminal sides and standard position of angles Right-triangle trigonometr: sine, cosine, and tangent Unit-circle trigonometr: sine, cosine, and tangent Radian measure of angles Angle measure conversion Sine and cosine functions of a general angle. Because the number 360 has so man integer divisors, man fractional revolutions have nice epressions when epressed as angles measured in degrees. Complete the table of fractional rotations or full rotations and their associated degree measures.. There are man easil epressed fractions of a full turn, and the measure of these epressions in radians is eas to determine. Complete the table of fractional rotations or full rotations and their associated radian measures. Table for Activit Rotation (in turns) Angle measure (in degrees) Rotation (in turns) Angle measure (in degrees) Rotation (in turns) 3 60 Angle measure (in degrees) Table for Activit Rotation (in turns) Angle measure (in radians) 4 3 Rotation (in turns) Angle measure (in radians) Rotation (in turns) 3 60 Angle measure (in radians) Copright Houghton Mifflin Compan. All rights reserved.

20 A0 APPENDIX A 3. Convert the following angles in degrees into angles in radians, and sketch the angles on a unit circle. a. 0 b. 700 c. 90 d Convert the following angles in radians to angles in degrees, and sketch the angles on a unit circle. 3 7 a. b. c. 6 d Ccling Superbike II, the $5,000 bab of the U.S. Ccling Federation (USCF), made its debut in the 996 Olmpics. Superbike II s front wheel is 3.6 inches in diameter, and its rear wheel is 7.56 inches in diameter. Over a distance of mile, how man turns does each wheel make? 6. Auto Wheels Use degree measure to write the angle that the tenths wheel on the odometer turns when the drive shaft of the automobile in Eample 6 completes one turn. 7. Ferris Wheel Consider a Ferris wheel on which are 30 equall spaced seats. a. Through what angle does the wheel move between stops to release passengers in two consecutive seats? b. What is the radian measurement of this angle? c. If the Ferris wheel is 00 feet in diameter, what is the distance traveled b each of the seats as the operator stops between consecutive seats to echange passengers?. Auto Wheels Suppose the owner of the automobile in Eample 6 replaced the tires with oversized tires that were inches in diameter. a. What would be the error created on the odometer readings? b. If the tires were replaced with tires that have an -inch diameter, what would be the error caused on the odometer? c. If the regular tires (0-inch diameter) lost 0.00 inch of their diameter through wear, how would the odometer readings be affected? 9. For finer measurements, the degree is further divided into 60 parts, and each such part is called a minute. The smbol used to indicate minutes is the single prime; for eample, 5 minutes is epressed as 0 5. An angle consisting of 00 minutes could also be epressed as 3 0. For even finer measurements, the minute is further divided into 60 equal parts called seconds. The smbol for a second is the double prime, so an angle of 30 degrees 4 minutes 7 seconds would be written as You might tr to estimate how small a second is to realize how much precision is used when some jobs state acceptable tolerances in terms of a seconds of arc. Convert into decimal degree equivalents the following angles given in degrees, minutes, and seconds. (Note: The Saturn missile used in the moon shots had guidance computers placed on a stable gimbaled platform that had to sta within 4 seconds of arc for the first 5 minutes of the launch.) a b Refer to the definitions in Activit 9, and convert into measures using degrees, minutes, and seconds the following angles given in decimal degrees. a b Suppose ou were to divide a rotation into 776 pieces. We will call each piece a patriotic unit and will use the abbreviation PU. a. How man rotations does each of the following patriotic unit measurements represent? i. 776 PU ii. 33 PU iii. PU iv. 444 PU v. PU vi. PU b. Convert each of the patriotic unit measurements in part a to radians. c. Find the value of each of the following: i. cos(776 PU) ii. sin(776 PU) iii. cos(666 PU) iv. sin(666 PU). Clock Creating different angle measurement sstems such as degrees and grads for partial turns and calculating their conversion factors is like making a 5-hour clock. Mr. Morton Rachofsk has built a 5-hour clock for the Circadian Clock Compan. This clock divides the 6,400 seconds in the standard da into 5 equal-length periods called hours. Noon on the clock is the same as noon on our regular time scale, but the other hour marks are different. Scientists conducting eperiments in the 930s observed people in caves where the could not see the sun. These people developed activit ccles that lasted 5 hours. Because we cannot change the solar da, Mr. Rachofsk said, Wh not change the clock? (Source: New York Times. October 7, 996, p. 47.) Copright Houghton Mifflin Compan. All rights reserved.

21 Trigonometr Basics A a. How long are Mr. Rachofsk hours in our regular minutes? b. What time on Mr. Rachofsk s clock is the regular clock time 3:00 P.M.? c. What is the regular clock time when it is 6:00 P.M. on Mr. Rachofsk s clock? For Activities 3 through, identif the trigonometric function ou would use to find the length of the indicated side. Assume that the triangle given is a right triangle with one of the other two angles identified as angle. 3. Given the length of the hpotenuse, find the length of the leg adjacent to angle. 4. Given the length of the leg opposite angle, find the length of the leg adjacent to angle. 5. Find the length of the leg opposite angle, given the length of the leg adjacent to angle. 6. Find the length of the hpotenuse, given the length of the leg opposite angle. 7. Given the length of the leg adjacent to angle, find the length of the hpotenuse.. One angle of the triangle is The leg opposite this angle is 5 inches long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the hpotenuse. 3. One angle of the triangle is 5.4. The leg adjacent to this angle is.5 miles long. a. Find the length of the hpotenuse. b. Find the length of the leg opposite the given angle. 4. One angle of the triangle is 0.. The hpotenuse is 3 centimeters long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the leg opposite the given angle. 5. Roof A gable is the triangular segment of a wall created b the roof. The pitch of a gable is its height divided b its width. See figure. (That is, pitch is equal to half the slope of the roof.) Consider a gable 40 feet wide feet with an angle of 30 at its peak.. Given the length of the hpotenuse, find the length of the leg opposite angle. For Activities 9 through 4, solve for the length of the indicated side. Assume that the triangle given is a right triangle. Height Width 9. One angle of the triangle is 0. The leg opposite this angle is 5 inches long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the hpotenuse. 0. One angle of the triangle is 7. The leg adjacent to this angle is meter long. a. Find the length of the hpotenuse. b. Find the length of the leg opposite the given angle.. One angle of the triangle is 5.. The hpotenuse is centimeters long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the leg opposite the given angle. a. How tall is the attic at its center? b. What is the pitch of the gable? c. How much board would be needed to cover the roof (not including the overhanging portion of the roof) if the house is 40 feet long? 6. Nuts Find the diameter of the smallest iron rod from which a heagonal nut with side 4mm can be cut. (Hint: The angle between two adjacent sides of the nut is 0.) 7. Stairwa A stairwa is to be constructed on a hill with a 34 incline. a. If each step is to have a 7-inch rise, what must be its tread (horizontal depth)? Copright Houghton Mifflin Compan. All rights reserved.

22 A APPENDIX A b. How man 7-inch steps will be needed if the length of the hill (measured up the slope) is 4 feet inches?. Navigation A boat that is sailing S 47 E is sailing on a trajector that is along an angle 47 east of true south. A ship sails. nautical miles S 47 E from its starting position. a. How far south has the ship sailed? b. How far east has it sailed? 9. Navigation Air Force pilots mark their bearing as the clockwise angle measured from the north. For eample, a bearing of 90 is east and a bearing of 0 is south. There is a landing strip 5 miles awa from a plane at bearing 33. a. What bearing is west? b. In what direction must the plane fl to reach the landing strip? c. If the plane were to fl directl north or south and then east or west to reach the landing strip, how far in each direction would the plane have to fl? 30. Because the sum of the three angles of an triangle is 0, a right triangle with one 45 angle must contain another 45 angle. As shown, this tpe of triangle is formed b two sides and a diagonal of a square. Thus the two legs of the triangle are of equal length. B the Pthagorean Theorem, the hpotenuse must have length s s s s 45 s s Using the lengths s and s, determine the eact values of the following trigonometric ratios. a. sin 45 b. cos 45 c. tan Use the values for sin 45 and cos 45 to determine the sine, cosine, and tangent values of each of the following angles. a. 35 b. 35 c s For Activities 3 through 39, sketch the given angle on the unit circle, and draw the appropriate right triangle corresponding to this angle. Calculate the sine and cosine of the angle, and indicate these values appropriatel on the sketch of the triangle Calculate sin 0 and cos 0 and eplain our answers in terms of the unit circle. Do the same thing for sin 90 and cos a. Use the graph to estimate sin 4, cos 4, sin, and cos. = 4 = b. Mark the estimated locations of the angles 44 and 6 on the figure, and use the grid to estimate the values of sin(), cos(), sin , and cos a. Use the graph to estimate sin.5, cos.5, 5 5 sin 6, and cos 6. b. Mark the estimated locations of the angles 5 and on the figure, and use the grid to estimate the values of sin, cos, 5 5 sin, and cos. Copright Houghton Mifflin Compan. All rights reserved.

23 Trigonometr Basics A3 49. a. Without using technolog, indicate whether each of the following values is positive, negative, or zero b using its interpretation as the -coordinate of a point on the unit circle. =.5 = 5 6 Trig value sin 0 3 sin 4 sin sin 4 sin Sign Trig value sin 4 sin sin 4 sin Sign 43. a. Find the values of sin 3 and cos 3. b. Interpret each answer to part a in terms of the coordinates of a point on the unit circle. c. Give two other angles whose sine and cosine values are the same as those for a. Find the values of sin 5 and cos 5. b. Interpret each answer to part a in terms of the coordinates of a point on the unit circle. c. Give two other angles whose sine and cosine values are the same as those for a. Find the values of sin and cos. b. Interpret the answers to part a in terms of the unit circle. c. Give three other angles whose sine and cosine 3 values are the same as those for a. Find the values of sin 9 and cos 9. b. Interpret the answers to part a in terms of the unit circle. c. Give three other angles whose sine and cosine 4 values are the same as those for Is it possible for the trigonometric functions of an angle to be cos 0.5 and sin 0.5? Eplain. 4. Is it possible for the trigonometric functions of an angle to be cos 0.35 and sin 0.? Eplain. b. Use our calculator or computer to complete the following table. Compare the signs of the values to our results from part a. Trig value sin 0 3 sin 4 sin sin 4 sin Decimal value Trig value sin 4 sin sin 4 sin Decimal value c. Plot the values ou obtained in part b as a function of the angle. How is this plot related to the graph of the function f() sin? 50. a. Without using technolog, indicate whether each of the following values is positive, negative, or zero b using its interpretation as the - coordinate of a point on the unit circle. Trig value cos 0 3 cos 4 cos cos 4 cos Sign Trig value cos 4 cos cos 4 cos Sign b. Use technolog to complete the following table. Compare the signs of the values to our results from part a. Trig value cos 0 3 cos 4 cos cos 4 cos Decimal value Trig value cos 4 cos cos 4 cos Decimal value Copright Houghton Mifflin Compan. All rights reserved.

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25 Answers to Appendi A A5 Answers to Appendi A TRIGONOMETRY BASICS 5 3. Rotation (in turns):, 6,, 4,, Angle measure (in degrees): 30, 90, 35, 360, 00,, a. radians π 35 b. 9 radians c. radians 35π 9 π _ b..643 miles 5. a feet b c square feet 7. a. 0.3 inches b. Four steps 9. a. Bearing 70 b. Northwest c miles north and.347 miles west 3. sin cos tan 45 a. 35 b. 35 c. 5 sin cos d radian The front wheel makes approimatel 53.6 turns in mile; the rear wheel makes approimatel turns in mile. 7. a. b. 5 radian c. Approimatel 0.47 feet 9. a. Approimatel 5.50 b. Approimatel a. i. rotation ii. 4 rotation iii. rotation iv. 4 rotation v. rotation vi. 6 rotation b. i. radians ii..5 radians iii. radians iv. radians v. 4 radians vi. radians c. i. ii. 0 iii iv cos 5. tan 7. cos 9. a inches b inches. a..50 centimeters b centimeters 3. a..703 miles Copright Houghton Mifflin Compan. All rights reserved.

26 A6 APPENDIX A = 6 = a. sin 4 0., cos 4 0.7, sin 0.4, cos 0.9 (Answers ma var.) 44 b. sin () 0., cos() 0.5, sin 6 0, 44 cos 6 (Answers ma var.) 43. a. sin , cos b. The point on the unit circle corresponding to an angle of 3 is approimatel (0.5, 0.66). 4 c. Two possible answers are 3 and a. sin, cos 0 b. The point on the unit circle corresponding to an angle 3 of is (0, ). 7 c. Possible answers include,, and. 47. It is not possible to have an angle such that cos 0.5 and sin 0.5 because cos and sin are - and - coordinates on the unit circle and must satisf the equation cos sin. 49. a. Zero, positive, positive, positive, zero Negative, negative, negative, zero b. 0, 0.707,, 0.707, ,, 0.707, 0 Copright Houghton Mifflin Compan. All rights reserved.