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All Textbook Solutions for Precalculus
39PEFor Exercises 37-40, convert from degrees to radians. Give the answers in exact form in terms of . (See Example 3) 19541PE42PEFor Exercises 41-44, convert from degrees to radians. Round to 4 decimal places. 12636For Exercises 41-44, convert from degrees to radians. Round to 4 decimal places. 180429For Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 4For Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 11647PEFor Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 7649PE50PEFor Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 2552PE53PEFor Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 5.3For Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 92For Exercises 45-56, convert from radians to decimal degrees. Round to 1 decimal place if necessary. (See Example 4) 757PEFor Exercises 57-64, find a positive angle and a negative angle that is coterminal to the given angle. (See Examples 5 and 6) 313For Exercises 57-64, find a positive angle and a negative angle that is coterminal to the given angle. (See Examples 5 and 6) 105For Exercises 57-64, find a positive angle and a negative angle that is coterminal to the given angle. (See Examples 5 and 6) 12For Exercises 57-64, find a positive angle and a negative angle that is coterminal to the given angle. (See Examples 5 and 6) 56For Exercises 57-64, find a positive angle and a negative angle that is coterminal to the given angle. (See Examples 5 and 6) 3463PEFor Exercises 57-64, find a positive angle and a negative angle that is coterminal to the given angle. (See Examples 5 and 6)For Exercises 65-70, find an angle between 0 and 360 or between 0 and 2 that is coterminal to the given angle. (See Examples 5 and 6) 1521For Exercises 65-70, find an angle between 0 and 360 or between 0 and 2 that is coterminal to the given angle. (See Examples 5 and 6) 603For Exercises 65-70, find an angle between 0 and 360 or between 0 and 2 that is coterminal to the given angle. (See Examples 5 and 6) 174For Exercises 65-70, find an angle between 0 and 360 or between 0 and 2 that is coterminal to the given angle. (See examples 5 and 6) 113For Exercises 65-70, find an angle between 0 and 360 or between 0 and 2 that is coterminal to the given angle. (See Examples 5 and 6) 718For Exercises 65-70, find an angle between 0 and 360 or between 0 and 2 that is coterminal to the given angle. (See Examples 5 and 6) 5971PEFor Exercises 71-74, find the exact length of the arc intercepted by a central angle on a circle of radius r . Then round to the nearest tenth of a unit. (See Example 7) =56,r=4m73PEFor Exercises 71-74, find the exact length of the arc intercepted by a central angle on a circle of radius r . Then round to the nearest tenth of a unit. (See Example 7) =135,r=2ydA 6-ft pendulum swings through an angle of 4036 . What is the length of the arc that the tip of the pendulum travels? Round to the nearest hundredth of a foot.A gear with a 1.2-cm radius moves through an angle of 22015 . What distance does a point on the edge of the gear move? Round to the nearest tenth of a centimeter.a. What is the geographical relationship between two points that have the same latitude? b. What is the geographical relationship between two points that have the same longitude?For Exercises 79-82, assume that the Earth is approximately spherical with radius 3960mi . Approximate the distances to the nearest mile. (See Example 8) Barrow, Alaska 71.3N,156.8W and Kailua, Hawaii 19.7N,156.1W have approximately the same longitude, which means that they are roughly due north-south of each other. Use the difference in latitude to approximate the distance between the cities.For Exercises 79-82, assume that the Earth is approximately spherical with radius 3960mi . Approximate the distances to the nearest mile. (See Example 8) Rochester, New York 43.2N,77.6W , and Richmond, Virginia 37.5N,77.5W , have approximately the same longitude, which means that they are roughly due north-south of each other. Use the difference in latitude to approximate the distance between the cities.For Exercises 79-82, assume that the Earth is approximately spherical with radius 3960mi . Approximate the distances to the nearest mile. (See Example 8) Raleigh, North Carolina35.8N,78.6W , is located north of the equator, and Quito, Ecuador 0.3S,78.6W , is located south of the equator. The longitudes are the same indicating that the cities are due north-south of each other. Use the difference in latitude to approximate the distance between the cities.For Exercises 79-82, assume that the Earth is approximately spherical with radius 3960mi . Approximate the distances to the nearest mile. (See Example 8) Trenton, New Jersey 40.2N,74.8W , is located north of the equator, and Ayacucho, Peru 13.2S,74.2W is located south of the equator. The longitudes are nearly the same indicating that the cities are roughly due north-south of each other. Use the difference in latitude to approximate the distance between the cities.A pulley is 16cm in diameter. a. Find the distance the load will rise if the pulley is rotated 1350 . Find the exact distance in terms of and then round to the nearest centimeter. b. Through how many degrees should the pulley rotate to lift the load 100cm ? Round to the nearest degree.A pulley is 1.2ft. in diameter. a. Find the distance the load will rise if the pulley is rotated 630 . Find the exact distance in terms of 8 and then round to the nearest tenth of a foot. b. Through how many degrees should the pulley rotate to lift the load 24ft ? Round to the nearest degree.A hoist is used to lift a palette of bricks. The drum on the hoist is 15in. in diameter. How many degrees should the drum be rotated to lift the palette a distance of 6ft ? Round to the nearest degree.A winch on a sailboat is 8in. in diameter and is used to pull in the "sheets" (ropes used to control the corners of a sail). To the nearest degree, how far should the winch be turned to pull in 2ft of rope? Before the widespread introduction of electronic devices to measure distances, surveyors used a subtense bar to measure a distance x that is not directly measurable. A subtense bar is a bar of known length h with marks or “targets" at either end. The surveyor measures the angle formed by the location of the surveyor's scope and the top and bottom of the bar (this is the angle subtended by the bar). Since the angle and height of the bar are known, right triangle trigonometry can be used to find the horizontal distance. Alternatively, if the distance from the surveyor to the bar is large, then the distance can be approximated by the radius r of the arc s intercepted by the bar. Use this information for Exercises 87-88.A surveyor uses a subtense bar to find the distance across a river. If the angle of sight between the bottom and top marks on a 2-m bar is 5718 , approximate the distance across the river between the surveyor and the bar. Round to the nearest meter.A surveyor uses a subtense bar to find the distance across a canyon. If the angle of sight between the bottom and top marks on a 2-m bar is 2433 , approximate the distance across the river between the surveyor and the bar. Round to the nearest meter.A circular paddle wheel of radius 3ft is lowered into a flowing river. The current causes the wheel to rotate at a speed of 12rpm . To 1 decimal place, a. What is the angular speed? (See Example 9) b. Find the speed of the current in ft/min. c. Find the speed of the current in mph.An energy-efficient hard drive has a 2.5-in . diameter and spins at 4200rpm . a. What is the angular speed? b. How fast in in./min does a point on the edge of the hard drive spin? Give the exact speed and the speed rounded to the nearest in./minA 714-in.-diameter circular saw has 24 teeth and spins at 5800rpm . a. What is the angular speed? b. What is the linear speed of one of the "teeth" on the outer edge of the blade? Round to the nearest inch per minute.On a weed-cutting device, a thick nylon line rotates on a spindle at 3000rpm . a. Determine the angular speed. b. Determine the linear speed (to the nearest inch per minute) of a point on the tip of the line if the line is 5in .A truck has 2.5-ft tires (in diameter). a. What distance will the truck travel with one rotation of the wheels? Give the exact distance and an approximation to the nearest tenth of a foot. b. How far will the truck travel with 10,000 rotations of the wheels? Give the exact distance and an approximation to the nearest foot. c. If the wheels turn at 672rpm , what is the angular speed? d. If the wheels turn at 672rpm , what is the linear speed in feet per minute? Give the exact distance and an approximation to the nearest whole unit. e. If the wheels turn at 672rpm , what is the linear speed in miles per hour? Round to the nearest mile per hour.A bicycle has 25-in . wheels (in diameter). a. What distance will the bicycle travel with one rotation of the wheels? Give the exact distance and an approximation to the nearest tenth of an inch. b. How far will the bicycle travel with 200 rotations of the wheels? Give the exact distance and approximations to the nearest inch and nearest foot. c. If the wheels turn at 80rpm , what is the angular speed? d. If the wheels turn at 80rpm , what is the linear speed in inches per minute? Give the exact speed and an approximation to the nearest inch per minute. e. If the wheels turn at 80rpm , what is the linear speed in miles per hours? Round to the nearest mile per hour.95PEFor Exercises 95-98, find the exact area of the sector. Then round the result to the nearest tenth of a unit. (See Example 10)97PEFor Exercises 95-98, find the exact area of the sector. Then round the result to the nearest tenth of a unit. (See Example 10)A slice of a circular pizza 12in .in diameter is cut into a wedge with a 45 angle. Find the area and round to the nearest tenth of a square inch.A circular cheesecake 9in. in diameter is cut into a slice with a 20 angle. Find the area and round to the nearest tenth of a square inch.The back wiper blade on an SUV extends 3in. from the pivot point to a distance of 17in. from the pivot point. If the blade rotates through an angle of 175 , how much area does it cover? Round to the nearest square inch.A robotic arm rotates through an angle of 160 . It sprays paint between a distance of 0.5 ft and 3 ft from the pivot point. Determine the amount of area that the arm makes. Round to the nearest square foot.103PE104PE105PE106PE107PE108PEThe second hand of a clock moves from 12:10 to 12:30 . a. How many degrees does it move during this time? b. How many radians does it move during this time? c. If the second hand is 10in. in length, determine the exact distance that the tip of the second hand travels during this time. d. Determine the exact angular speed of the second hand in radians per second. e. What is the exact linear speed (in inches per second) of the tip of the second hand? f. What is the amount of area that the second hand sweeps out during this time? Give the exact area in terms of and then approximate to the nearest square inchThe minute hand of a clock moves from 12:10 to 12:15 . a. How many degrees does it move during this time? b. How many radians does it move during this time? c. If the minute hand is 9in. in length, determine the exact distance that the tip of the minute hand travels during this time. d. Determine the exact angular speed of the minute hand in radians per minute. e. What is the exact linear speed (in inches per minute) of the tip of the minute hand? f. What is the amount of area that the minute hand sweeps out during this time? Give the exact area in terms of and then approximate to the nearest square inch.The Earth's orbit around the Sun is elliptical (oval shaped). However, the elongation is small, and for our discussion here, we take the orbit to be circular with a radius of approximately 93,000,000mi . a. Find the linear speed (in mph) of the Earth through its orbit around the Sun. Round to the nearest hundred miles per hour. b. How far does the Earth travel in its orbit in one day? Round to the nearest thousand miles.The Earth completes one full rotation around its axis (poles) each day. a. Determine the angular speed (in radians per hour) of the Earth during its rotation around its axis. b. The Earth is nearly spherical with a radius of approximately 3960 mi. Find the linear speed of a point on the surface of the Earth rounded to the nearest mile per hour.Two gears are calibrated so that the smaller gear drives the larger gear. For each rotation of the smaller gear, how many degrees will the larger gear rotate?Two gears are calibrated so that the larger gear drives the smaller gear. The larger gear has a 6-in . radius, and the smaller gear has a 1.5-in. radius. For each rotation of the larger gear, by how many degrees will the smaller gear rotate?A spinning-disc confocal microscope contains a rotating disk with multiple small holes arranged in a series of nested Archimedean spirals. An intense beam of light is projected through the holes, enabling biomedical researchers to obtain detailed video images of live cells. The spinning disk has a diameter of 55mm and rotates at a rate of 1800rpm . At the edge of the disk. a. Find the angular speed. b. Find the linear speed. Round to the nearest whole unit.For Exercises 116-119, approximate the area of the shaded region to 1 decimal place. In the figure, s represents arc length, and r represents the radius of the circle.For Exercises 116-119, approximate the area of the shaded region to 1 decimal place. In the figure, s represents arc length, and r represents the radius of the circle.For Exercises 116-119, approximate the area of the shaded region to 1 decimal place. In the figure, s represents arc length, and r represents the radius of the circle.For Exercises 116-119, approximate the area of the shaded region to 1 decimal place. In the figure, s represents arc length, and r represents the radius of the circle.Explain what is meant by 1 radian. Explain what is meant by 1 .For an angle drawn in standard position, explain how to determine in which quadrant the terminal side lines.As the fan rotates (see figure), which point A or B has a greater angular speed? Which point has a greater linear speed? Why?If an angle of a sector is held constant, but the radius is doubled, how will the arc length of the sector and area of the sector be affected?If an angle of a sector is doubled, but the radius is held constant, how will the arc length of the sector and the area of the sector be affected?When a person pedals a bicycle, the front sprocket moves a chain that drives the back wheel and propels the bicycle forward. For each rotation of the front sprocket the chain moves a distance equal to the circumference of the front sprocket. The back sprocket is smaller, so it will simultaneously move through a greater rotation. Furthermore, since the back sprocket is rigidly connected to the back wheel, each rotation of the back sprocket generates a rotation of the wheel. Suppose that the front sprocket of a bicycle has a 4-in . radius and the back sprocket has a 2-in . radius. a. How much chain will move with one rotation of the pedals (front sprocket)? b. How many times will the back sprocket turn with one rotation of the pedals? c. How many times will the wheels turn with one rotation of the pedals? d. If the wheels are 27in. in diameter, how far will the bicycle travel with one rotation of the pedals? e. If the bicyclist pedals 80rpm , what is the linear speed (in ft/min) of the bicycle? f. If the bicyclist pedals 80rpm , what is the linear speed (in mph) of the bicycle?In the third century B.C. , the Greek astronomer Eratosthenes approximated the Earth's circumference. On the summer solstice at noon in Alexandria, Egypt, Eratosthenes measured the angle of the Sun relative to a line perpendicular to the ground. At the same time in Siena (now Aswan), located on the Tropic of Cancer, the Sun was directly overhead. a. If =150 of a circle, find the measure of in degrees. (In Eratosthenes' time, the degree measure had not yet been defined.) b. If the distance between Alexandria and Siena is 5000 stadia, find the circumference of the Earth measured in stadia. c. If 10stadia1mi , find Eratosthenes' approximation of the circumference of the Earth in miles (the modern-day approximation at the equator is 24,900mi ).The space shuttle program involved 135 manned space flights in 30yr . In addition to supplying and transporting astronauts to the International Space Station, space shuttle missions serviced the Hubble Space Telescope and deployed satellites. For a particular mission, a space shuttle orbited the Earth in 1.5hr . at an altitude of 200mi . a. Determine the angular speed (in radians per hour) of the shuttle. b. Determine the linear speed of the shuttle in miles per hour. Assume that the Earth's radius is 3960mi . Round to the nearest hundred miles per hour128PE129PE130PE131PE132PE133PE134PESuppose that the real number t corresponds to the point P55,255 on the unit circle. Evaluate the six trigonometric functions of t .2SP3SPGiven sint=37 and cost=2107 use the reciprocal and quotient identities to find the values of the other trigonometric functions of t .5SP6SP7SPUse the properties of the trigonometric functions to simplify. a. sect2sect b. sin2t+2sin2tUse a calculator to approximate the function values. Round to 4 decimal places. a. tan1.4 b. cot8The graph of x2+y2=1 is known as the circle. It has a radius of length and center at the .The circumference of the unit circle is . Thus, the t=2 value represents of a revolution.3PE4PE5PEThe domain of the trigonometric functions and is all real numbers.7PE8PEThe period of the tangent and cotangent functions is . The period of the sine, cosine, cosecant, and secant functions is .The cosine function is an function because cost=cost . The sine function is an function because sint=sint .For Exercises 11-14, determine if the point lies on the unit circle. 105,15512PE13PEFor Exercises 11-14, determine if the point lies on the unit circle. 23417,31717For Exercises 15-18, the real number t corresponds to the point P on the unit circle. Evaluate the six trigonometric functions of t . (See Example 1)For Exercises 15-18, the real number t corresponds to the point P on the unit circle. Evaluate the six trigonometric functions of t . (See Example 1)17PE18PE19PE20PEFor Exercises 20-23, identify the coordinates of point P . Then evaluate the six trigonometric functions of t . (See Example 2)For Exercises 20-23, identify the coordinates of point P . Then evaluate the six trigonometric functions of t . (See Example 2)23PEFor Exercises 24-26, identify the ordered pairs on the unit circle corresponding to each real number t . a. t=23 b. t=54 c. t=5625PEFor Exercises 24-26, identify the ordered pairs on the unit circle corresponding to each real number t a. t=53 b. t=74 c. t=6For Exercises 27-32, evaluate the trigonometric function at the given real number. ft=cost;t=23For Exercises 27-32, evaluate the trigonometric function at the given real number. gt=sint;t=54For Exercises 27-32, evaluate the trigonometric function at the given real number. ht=cott;t=116For Exercises 27-32, evaluate the trigonometric function at the given real number. st=sect;t=53For Exercises 27-32, evaluate the trigonometric function at the given real number. zt=csct;t=74For Exercises 27-32, evaluate the trigonometric function at the given real number. rt=tant;t=56For Exercises 33-38, select the domain of the trigonometric function. a. All real numbers b. tt2n+12forallintegersn c. ttnforallintegersn ft=sint34PEFor Exercises 33-38, select the domain of the trigonometric function a. All real numbers b. tt2n+12forallintegersn c. ttnforallintegersn ft=cottFor Exercises 33-38, select the domain of the trigonometric function a. All real numbers b. tt2n+12forallintegersn c. ttnforallintegersn ft=cost37PE38PEFor Exercises 39-42, evaluate the function if possible. (See Example 3) a. sin0 b. cot c. tan3 d. sec e. csc0 f. cosFor Exercises 39-42, evaluate the function if possible. (See Example 3) a. cos2 b. csc2 c. cot52 d. tan2 e. sec32 f. sin2For Exercises 39-42, evaluate the function if possible. (See Example 3) a. sin32 b. cos72 c. tan32 d. csc2 e. sec1.5 f. cot2For Exercises 39-42, evaluate the function if possible. (See Example 3) a. cot0 b. cos2 c. csc d. tan0 e. sin3 f. sec0For Exercises 43-46, given the values for sint and cost , use the reciprocal and quotient identities to find the values of the other trigonometric functions of t . (See Example 4) sint=53 and cost=23For Exercises 43-46, given the values for sint and cost , use the reciprocal and quotient identities to find the values of the other trigonometric functions of t . (See Example 4) sint=34 and cost=7445PEFor Exercises 43-46, given the values for sint and cost , use the reciprocal and quotient identities to find the values of the other trigonometric functions of t . (See Example 4) sint=2853 and cost=4553For Exercises 47-48, derive the given identity from the Pythagorean identity, sin2t+cos2t=1 . tan2t+1=sec2t48PEFor Exercises 49-54, use an appropriate Pythagorean identity to find the indicated value. (See Example 5) Given cost=725 for 2t , find the value of sint .For Exercises 49-54, use an appropriate Pythagorean identity to find the indicated value. (See Example 5) Given sint=817 for t32 , find the value of cost .For Exercises 49-54, use an appropriate Pythagorean identity to find the indicated value. (See Example 5) Given cott=4528 for t32 , find the value ofcsct .For Exercises 49-54, use an appropriate Pythagorean identity to find the indicated value. (See Example 5) Given csct=4140 for 32t2 , find the value ofcott .53PE54PEWrite sin t in terms of cos t for a. t in Quadrant I. (See Example 6) b. t in Quadrant III.56PEWrite cott in terms of csct for a. t in Quadrant I. b. t in Quadrant III.Write cost in terms of sint for a. t in Quadrant II. b. t in Quadrant IVGiven that cos2912=624 determine the value of cos512 (See Example 7)Given that sec1112=26 , determine the value of sec131261PEGiven that sin10=514 , determine the value of csc2110 .For Exercises 63-66, use the periodic properties of the trigonometric functions to simplify each expression to a single function of t . sint+2cott+For Exercises 63-66, use the periodic properties of the trigonometric functions to simplify each expression to a single function of t . sint+2sect+265PE66PEFor Exercises 67-74, use the even-odd and periodic properties of the trigonometric functions to simplify. (See Example 8) csct4csct68PEFor Exercises 67-74, use the even-odd and periodic properties of the trigonometric functions to simplify. (See Example 8) cott+cottFor Exercises 67-74, use the even odd and periodic properties of the trigonometric functions to simplify. (See Example 8) sect+2sect71PEFor Exercises 67-74, use the even-odd and periodic properties of the trigonometric functions to simplify. (See Example 8) cot3t3cot3t+For Exercises 67-74, use the even-odd and periodic properties of the trigonometric functions to simplify. (See Example 8) cos2tcos2tFor Exercises 67-74, use the even-odd and periodic properties of the trigonometric functions to simplify. (See Example 8) sec2t+3sec2tUse a calculator to approximate the function values. Round to 4 decimal places. (See Example 9) a sin0.15 b. cos25Use a calculator to approximate the function values. Round to 4 decimal places. (See Example 9) a. cos711 b. sin0.96Use a calculator to approximate the function values. Round to 4 decimal places. (See Example 9) a. cot127 b. sec5.43Use a calculator to approximate the function values. Round to 4 decimal places. (See Example 9) a. csc7.58 b. tan38For Exercises 79-80 evaluate sint, cost, and tant for the real number t . a. t=23 b. t=4380PE81PEFor Exercises 81-86, identify the values of t on the interval 0,2 that make the function undefined (if any). y=cottFor Exercises 81-86, identify the values of t on the interval 0,2 that make the function undefined (if any). y=tant84PEFor Exercises 81-86, identify the values of t on the interval 0,2 that make the function undefined (if any). y=csctFor Exercises 81-86, identify the values of t on the interval 0,2 that make the function undefined (if any). y=sectFor Exercises 87-92, select all properties that apply to the trigonometric function. a. The function is even. b. the function is odd. c. The period is 2 . d. The period is . e. The domain is all real numbers. f. The domain is all real numbers excluding odd multiples of 2 . g. The domain is all real number excluding multiples of . ft=sintFor Exercises 87-92, select all properties that apply to the trigonometric function. a. The function is even. b. the function is odd. c. The period is 2 . d. The period is . e. The domain is all real numbers. f. The domain is all real numbers excluding odd multiples of 2 . g. The domain is all real number excluding multiples of . ft=tantFor Exercises 87-92, select all properties that apply to the trigonometric function. a. The function is even. b. the function is odd. c. The period is 2 . d. The period is . e. The domain is all real numbers. f. The domain is all real numbers excluding odd multiples of 2 . g. The domain is all real number excluding multiples of . ft=sectFor Exercises 87-92, select all properties that apply to the trigonometric function. a. The function is even. b. the function is odd. c. The period is 2 . d. The period is . e. The domain is all real numbers. f. The domain is all real numbers excluding odd multiples of 2 . g. The domain is all real number excluding multiples of . ft=cottFor Exercises 87-92, select all properties that apply to the trigonometric function. a. The function is even. b. the function is odd c. The period is 2 d. The period is e. The domain is all real numbers. f. The domain is all real numbers excluding odd multiples of 2 . g. The domain is all real number excluding multiples of . ft=csctFor Exercises 87-92, select all properties that apply to the trigonometric function. a. The function is even. b. the function is odd. c. The period is 2 . d. The period is . e. The domain is all real numbers. f. The domain is all real numbers excluding odd multiples of 2 . g. The domain is all real number excluding multiples of . ft=cost93PE94PE95PEFor exercises 93-98, simplify using properties of trigonometric functions. sin26cos2697PEFor exercises 93-98, simplify using properties of trigonometric functions. tan223+csc25499PEThe fluctuating brightness of a distant star is given by the function fd=3.8+0.25sin23d Where d is the number of days and fd is the apparent brightness. Complete the table and give the period of the function. Round to 2 decimal places.101PEFor exercises 101-104, identify each function as even, odd, or neither. gt=tcost103PEFor exercises 101-104, identify each function as even, odd, or neither. ht=t3+sectDescribe the changes in sint and cost as t increases from 0 to 2106PEExplain why 1cost1 and 1sint1 for all real numbers t .108PEFor Exercises 109-114, use the figure to estimate the value of (a) sint and (b) cost for the given value of t . t=0.5110PE111PEFor Exercises 109-114, use the figure to estimate the value of (a) sint and (b) cost for the given value of t . t=2.75113PE114PEFor Exercises 115-120, use the figure from exercises 109-114 to approximate the solutions to the equation over the interval 0,2 . sint=0.2116PE117PEFor Exercises 115-120, use the figure from exercises 109-114 to approximate the solutions to the equation over the interval 0,2 . cost=0.6119PE120PE121PEProve that the period of ft=cost is 2 .123PEConsider the expression 1cost1 a. What is the value of the numerator for t=0 ? b. What is the value of the denominator for t=0 ? c. The expression 1costt is undefined at t=0 . Complete the table to investigate the value of the expression close to t=0 . Round to 5 decimal places.1SP2SP3SPEvaluate sin30 , cos30 , and tan30 .Simplify the expressions. a. cot60cot30 b. sin3cos6+cos3sin6For each function value, find a cofunction with the same value. a. tan22.5=21 b. sec3=2Suppose that a farm worker determines that the distance along the ground from her position to the base of a palm tree is 16ft . She measures the angle of elevation from an eye level of 5ft to the top of the tree as 45.9 . Is the tree tall enough to harvest (at least 20ft tall)?If a 15-ft ladder is leaning against a wall at an angle of 62 with the ground, how high up the wall will the ladder reach? Round to the nearest tenth of a foot.In a right triangle with an acute angle the longest side in the triangle is the and is opposite the angle.The leg of a right triangle that lies on one ray of angle is called the leg, and the leg that lies across the triangle from is called the leg.The mnemonic device "SOH-CAH-TOA" stands for the ratios sin=,cos=, and tan=Complete the reciprocal and quotient identities, csc=1sec=1cot=1orGiven the lengths of two sides of a right triangle, we can find the length of the third side by using the theorem.An right triangle is a right triangle in which the two legs are of equal length. The two acute angles in this triangle each measure .The length of the shorter leg of a 306090 triangle is always the length of the hypotenuse.For the six trigonometric functions sin,cos,tan,csc,sec, and cot, identify the three reciprocal pairs.tan is the of sin and cos .Complete the Pythagorean identities. sin2+=1,+1=sec2,1+cot2=The two acute angles in a right triangle are complementary because the sum of their measures is .The sine of angle equals the cosine of . For this reason, the sine and cosine functions are called .For Exercises 13-14, find the exact values of the six trigonometric functions for angle .For Exercises 13-14, find the exact values of the six trigonometric functions for angle .For Exercises 15-18, first use the Pythagorean theorem to find the length of the missing side. Then find the exact values of the six trigonometric functions for angle . (See Example 1)For Exercises 15-18, first use the Pythagorean theorem to find the length of the missing side. Then find the exact values of the six trigonometric functions for angle . (See Example 1)17PEFor Exercises 15-18, first use the Pythagorean theorem to find the length of the missing side. Then find the exact values of the six trigonometric functions for angle . (See Example 1)For Exercises 19-22, first use the Pythagorean theorem to find the length of the missing side of the right triangle. Then find the exact values of the six trigonometric functions for the angle opposite the shortest side. (See Example 1) Leg=2cm,leg=6cmFor Exercises 19-22, first use the Pythagorean theorem to find the length of the missing side of the right triangle. Then find the exact values of the six trigonometric functions for the angle opposite the shortest side. (See Example 1) Leg=3cm,leg=15cm21PE22PEIn Exercises 23-24, given the value of one trigonometric function of an acute angle , find the values of the remaining five trigonometric functions of . (See Example2) tan=47In Exercises 23-24, given the value of one trigonometric function of an acute angle , find the values of the remaining five trigonometric functions of . (See Example2) cos=1010For Exercises 25-30, assume that is an acute angle. (See Example 2) If cos=217 , find csc .For Exercises 25-30, assume that is an acute angle. (See Example 2) If sin=1717 , find cot .27PEFor Exercises 25-30, assume that is an acute angle. (See Example 2) If csc=3 , find cos .29PEFor Exercises 25-30, assume that is an acute angle. (See Example 2) If cot=32 , find cos .31PEa. Evaluate sin60 . b. Evaluate sin30+sin30 . c. Are the values in parts (a) and (b) the same?For Exercises 33-38, find the exact value of each expression without the use of a calculator. (See Example 5) sin4cot3For Exercises 33-38, find the exact value of each expression without the use of a calculator. (See Example 5) tan6csc435PEFor Exercises 33-38, find the exact value of each expression without the use of a calculator. (See Example 5) 2cos65sin3For Exercises 33-38, find the exact value of each expression without the use of a calculator. (See Example 5) csc260sin245For Exercises 33-38, find the exact value of each expression without the use of a calculator. (See Example 5) cos245+tan26039PE40PEFor Exercises 39-44, determine whether the statement is true or false for an acute angle by using the fundamental identities. If the statement is false, provide a counterexample by using a special angle: 3,4, or 6 . sin2+tan2+cos2=sec242PE43PEFor Exercises 39-44, determine whether the statement is true or false for an acute angle by using the fundamental identities. If the statement is false, provide a counterexample by using a special angle: 3,4, or 6 . sincostan+1=cos2For Exercises 45-50, given the function value, find a cofunction of another angle with the same value. (See Example 6) tan75=2+3For Exercises 45-50, given the function value, find a cofunction of another angle with the same value. (See Example 6) sec12=6247PEFor Exercises 45-50, given the function value, find a cofunction of another angle with the same value. (See Example 6) sin15=624For Exercises 45-50, given the function value, find a cofunction of another angle with the same value. (See Example 6) cos4=22For Exercises 45-50, given the function value, find a cofunction of another angle with the same value. (See Example 6) cot6=3For Exercises 51-54, use a calculator to approximate the function values to 4 decimal places. Be sure that your calculator is in the correct mode. a. cos48.2 b. sin25542 c. tan38For Exercises 51-54, use a calculator to approximate the function values to 4 decimal places. Be sure that your calculator is in the correct mode. a. sin12.6 b. tan193618 c. cos522For Exercises 51-54, use a calculator to approximate the function values to 4 decimal places. Be sure that your calculator is in the correct mode. a. csc39.84 b. sec18 c. cot0.8For Exercises 51-54, use a calculator to approximate the function values to 4 decimal places. Be sure that your calculator is in the correct mode. a. cot18.46 b csc29 c. sec1.25An observer at the top of a 426-ft cliff measures the angle of depression from the top of a cliff to a point on the ground to be 5 . What is the distance from the base of the cliff to the point on the ground? Round to the nearest foot. (See Example 7)A lamppost casts a shadow of 18ft when the angle of elevation of the Sun is 33.7 . How high is the lamppost? Round to the nearest foot.A 30-ft boat ramp makes a 7 angle with the water. What is the height of the ramp above the water at the ramp’s highest point? Round to the nearest tenth of a foot. (See Example 8)58PEThe Lookout Mountain Incline Railway, located in Chattanooga, Tennessee, is 4972ft long and runs up the side of the mountain at an average incline of 17 . What is the gain in altitude? Round to the nearest foot.A 12-ft ladder learning against a house makes a 64 angle with the ground. Will the ladder reach a window sill that is 10.5 ft up from the base of the house?According to National Football League (NFL) rules, all crossbars on goalposts must be 10ft from the ground. However, teams are allowed some freedom on how high the vertical posts on each end may extend, as long as they measure at least 30ft . A measurement on an NFL field taken 100yd from the goalposts yields an angle of 7.8 from the ground to the top of the posts. If the crossbar is 10ft from the ground, do the goalposts satisfy the NFL rules?A zip line is to be built between two towers labeled A and B across a wetland area. To approximate the distance of the zip line, a surveyor marks a third point C, a distance of 175 ft from one end of the zip line and perpendicular to the zip line. The measure of ABC is74.5 . How long is the zip line? Round to the nearest foot.To determine the width of a river from point A to point B , a surveyor walks downriver 50ft along a line perpendicular to AB to a new position at point C . The surveyor determines that the measure of ACB is 60 . Find the exact width of the river from point A to point B .For Exercises 64-68, use the fundamental trigonometric identities as needed. Given that sinx0.3746 , approximate the given function values. Round to 4 decimal places. a. cos90x b. cosx c. tanx d. sin90x e. cot90x f. cscx