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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49EPROVE: Identities Involving Inverse Trigonometric Functions (a) Graph the function and make a conjecture, and (b) prove that your conjecture is true. 50. y=tan1x+tan11x51EFor an object in simple harmonic motion with amplitude a and period 2/, find an equation that models the displacement y at time t if (a) y = 0 at time t = 0: y = __________. (b) y = a at time t = 0: y = __________.For an object in damped harmonic motion with initial amplitude a, period 2/, and damping constant c, find an equation that models the displacement y at time t if (a) y = 0 at time t = 0: y = __________. (b) y = a at time t = 0: y = __________.(a) For an object in harmonic motion modeled by y = A sin(kt b) the amplitude is __________, the period is __________, and the phase is __________. To find the horizontal shift, we factor out k to get y = __________. From this form of the equation we see that the horizontal shift is __________. (b) For an object in harmonic motion modeled by y = 5 sin(4t ) the amplitude is __________, the period is __________, the phase is __________, and the horizontal shift is __________.Objects A and B are in harmonic motion modeled by y = 3 sin(2t ) and y=3sin(2t2). The phase of A is __________, and the phase of B is __________. The phase difference is __________, so the objects are moving __________ (in phase/out of phase).5E6ESimple Harmonic Motion The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. 7. y = cos 0.3tSimple Harmonic Motion The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. 8. y = 2.4 sin 3.6tSimple Harmonic Motion The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. 9. y=0.25cos(1.5t3)Simple Harmonic Motion The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. 10. y=32sin(0.2t+1.4)11E12ESimple Harmonic Motion Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time t = 0. 13. amplitude 10 cm, period 3 sSimple Harmonic Motion Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time t = 0. 14. amplitude 24 ft, period 2 minSimple Harmonic Motion Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time t = 0. 15. amplitude 6 in., frequency 5/ HzSimple Harmonic Motion Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time t = 0. 16. amplitude 1.2 m, frequency 0.5 HzSimple Harmonic Motion Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time t = 0. 17. amplitude 60 ft, period 0.5 minSimple Harmonic Motion Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time t = 0. 18. amplitude 35 cm, period 8 s19E20E21E22EDamped Harmonic Motion An initial amplitude k, damping constant c, and frequency f or period p are given. (Recall that frequency and period are related by the equation f = 1/p.) (a) Find a function that models the damped harmonic motion. Use a function of the form y = kect cos t in Exercises 2124 and of the form y = kect sin t in Exercises 2528. (b) Graph the function. 23. k = 100, c = 0.05, p = 424E25E26E27E28EAmplitude, Period, Phase, and Horizontal Shift For each sine curve find the amplitude, period, phase, and horizontal shift. 29. y=5sin(2t2)30E31E32E33E34E35E36E37E38EA Bobbing Cork A cork floating in a lake is bobbing in simple harmonic motion. Its displacement above the bottom of the lake is modeled by y=0.2cos20t+8 where y is measured in meters and t is measured in minutes. (a) Find the frequency of the motion of the cork. (b) Sketch a graph of y. (c) Find the maximum displacement of the cork above the lake bottom.FM Radio Signals The carrier wave for an FM radio signal is modeled by the function y=asin(2(9.15107)t) where t is measured in seconds. Find the period and frequency of the carrier wave.Blood Pressure Each time your heart beats, your blood pressure increases, then decreases as the heart rests between beats. A certain persons blood pressure is modeled by the function p(t)=115+25sin(160t) where p(t) is the pressure (in mmHg) at time t, measured in minutes. (a) Find the amplitude, period, and frequency of p. (b) Sketch a graph of p. (c) If a person is exercising, his or her heart beats faster. How does this affect the period and frequency of p?Predator Population Model In a predator/prey model, the predator population is modeled by the function y=900cos2t+8000 where t is measured in years. (a) What is the maximum population? (b) Find the length of time between successive periods of maximum population.Mass-Spring System A mass attached to a spring is moving up and down in simple harmonic motion. The graph gives its displacement d(t) from equilibrium at time t. Express the function d in the form d(t) = a sin t.Tides The graph shows the variation of the water level relative to mean sea level in Commencement Bay at Tacoma, Washington, for a particular 24-h period. Assuming that this variation is modeled by simple harmonic motion, find an equation of the form y = a sin t that describes the variation in water level as a function of the number of hours after midnight.Tides The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12-h period the water starts at mean sea level, rises to 21 ft above, drops to 21 ft below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12-h period.Mass-Spring System A mass suspended from a spring is pulled down a distance of 2 ft from its rest position, as shown in the figure. The mass is released at time t = 0 and allowed to oscillate. If the mass returns to this position after 1 s, find an equation that describes its motion.Mass-Spring System A mass is suspended on a spring. The spring is compressed so that the mass is located 5 cm above its rest position. The mass is released at time t = 0 and allowed to oscillate. It is observed that the mass reaches its lowest point 12s after it is released. Find an equation that describes the motion of the mass.48EFerris Wheel A Ferris wheel has a radius of 10 m, and the bottom of the wheel passes 1 m above the ground. If the Ferris wheel makes one complete revolution every 20 s, find an equation that gives the height above the ground of a person on the Ferris wheel as a function of time.Cock Pendulum The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is 10. We know from physical principles that the angle between the pendulum and its rest position changes in simple harmonic fashion. Find an equation that describes the size of the angle as a function of lime. (Take t = 0 to be a time when the pendulum is vertical.)Variable Stars The variable star Zeta Gemini has a period of 10 days. The average brightness of the star is 3.8 magnitudes, and the maximum variation from the average is 0.2 magnitude. Assuming that the variation in brightness is simple harmonic, find an equation that gives the brightness of the star as a function of time.Variable Stars Astronomers believe that the radius of a variable star increases and decreases with the brightness of the star. The variable star Delta Cephei (Example 4) has an average radius of 20 million miles and changes by a maximum of 1.5 million miles from this average during a single pulsation. Find an equation that describes the radius of this star as a function of time.Biological Clocks Circadian rhythms are biological processes that oscillate with a period of approximately 24 h. That is, a circadian rhythm is an internal daily biological clock. Blood pressure appears to follow such a rhythm. For a certain individual the average resting blood pressure varies from a maximum of 100 mmHg at 2:00 p.m. to a minimum of 80 mmHg at 2:00 a.m. Find a sine function of the form f(t)=asin((tc))+b that models the blood pressure at time t, measured in hours from midnight.Electric Generator The armature in an electric generator is rotating at the rate of 100 revolutions per second (rps). If the maximum voltage produced is 310 V, find an equation that describes this variation in voltage. What is the RMS voltage? (See Example 6 and the margin note adjacent to it.)Electric Generator The graph shows an oscilloscope reading of the variation in voltage of an AC current produced by a simple generator. (a) Find the maximum voltage produced. (b) Find the frequency (cycles per second) of the generator. (c) How many revolutions per second does the armature in the generator make? (d) Find a formula that describes the variation in voltage as a function of time.Doppler Effect When a car with its horn blowing drives by an observer, the pitch of the horn seems higher as it approaches and lower as it recedes (see the figure below). This phenomenon is called the Doppler effect. If the sound source is moving at speed v relative to the observer and if the speed of sound is v0, then the perceived frequency f is related to the actual frequency f0 as follows. f=f0(v0v0v) We choose the minus sign if the source is moving toward the observer and the plus sign if it is moving away. Suppose that a car drives at 110 ft/s past a woman standing on the shoulder of a highway, blowing its horn, which has a frequency of 500 Hz. Assume that the speed of sound is 1130 ft/s. (This is the speed in dry air at 70F.) (a) What are the frequencies of the sounds that the woman hears as the car approaches her and as it moves away from her? (b) Let A be the amplitude of the sound. Find functions of the form y=Asint that model the perceived sound as the car approaches the woman and as it recedes.Motion of a Building A strong gust of wind strikes a tall building, causing it to sway back and forth in damped harmonic motion. The frequency of the oscillation is 0.5 cycle per second, and the damping constant is c = 0.9. Find an equation that describes the motion of the building. (Assume that k = 1, and take t = 0 to be the instant when the gust of wind strikes the building.)Shock Absorber When a car hits a certain bump on the road, a shock absorber on the car is compressed a distance of 6 in., then released (see the figure). The shock absorber vibrates in damped harmonic motion with a frequency of 2 cycles per second. The damping constant for this particular shock absorber is 2.8. (a) Find an equation that describes the displacement of the shock absorber from its rest position as a function of time. Take t = 0 to be the instant that the shock absorber is released. (b) How long does it take for the amplitude of the vibration to decrease to 0.5 in.?Tuning Fork A tuning fork is struck and oscillates in damped harmonic motion. The amplitude of the motion is measured, and 3 s later it is found that the amplitude has dropped to 14 of this value. Find the damping constant c for this tuning fork.Guitar String A guitar string is pulled at point P a distance of 3 cm above its rest position. It is then released and vibrates in damped harmonic motion with a frequency of 165 cycles per second. After 2 s, it is observed that the amplitude of the vibration at point P is 0.6 cm. (a) Find the damping constant c. (b) Find an equation that describes the position of point P above its rest position as a function of time. Take t = 0 to be the instant that the string is released.Two Fans Electric fans A and B have radius 1 ft and, when switched on, rotate counterclockwise at the rate of 100 revolutions per minute. Starting with the position shown in the figure, the fans are simultaneously switched on. (a) For each fan, find an equation that gives the height of the red dot (above the horizontal line shown) t minutes after the fans are switched on. (b) Are the fans rotating in phase? Through what angle should fan A be rotated counterclockwise in order that the two fans rotate in phase?Alternating Current Alternating current is produced when an armature rotates about its axis in a magnetic field, as shown in the figure. Generators A and B rotate counterclockwise at 60 Hz (cycles per second) and each generator produces a maximum of 50 V. The voltage for each generator is modeled by EA=50sin(120t)EB=50sin(120t54) (a) Find the voltage phase for each generator, and find the phase difference. (b) Are the generators producing voltage in phase? Through what angle should the armature in the second generator be rotated counterclockwise in order that the two generators produce voltage in phase?DISCUSS: Phases of Sine The phase of a sine curve y = sin(kt + b) represents a particular location on the graph of the sine function y = sin t. Specifically, when t = 0, we have y = sin b, and this corresponds to the point (b, sin b) on the graph of y = sin t. Observe that each point on the graph of y = sin t has different characteristics. For example, for t = /6, we have sint=12 and the values of sine are increasing, whereas at t = 5/6, we also have sint=12 but the values of sine are decreasing. So each point on the graph of sine corresponds to a different phase of a sine curve. Complete the descriptions for each label on the graph below.DISCUSS: Phases of the Moon During the course of a lunar cycle (about 1 month) the moon undergoes the familiar lunar phases. The phases of the moon are completely analogous to the phases of the sine function described in Exercise 63. The figure below shows some phases of the lunar cycle starting with a new moon waxing crescent moon first quarter moon and so on. The next to last phase shown is a waning crescent moon. Give similar descriptions for the other phases of the moon shown in the figure. What are some events on the earth that follow a monthly cycle and are in phase with the lunar cycle? What are some events that are out of phase with the lunar cycle?1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC9RCC10RCC(a) What is simple harmonic motion? (b) What is damped harmonic motion? (c) Give real-world examples of harmonic motion.12RCC13RCC1RE2REReference Number and Terminal Point A real number t is given. (a) Find the reference number for t. (b) Find the terminal point P(x, y) on the unit circle determined by t. (c) Find the six trigonometric functions of t. 3. t=234RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28REHorizontal Shifts A trigonometric function is given. (a) Find the amplitude, period, and horizontal shift of the function. (b) Sketch the graph. 29. y=10cos12x30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55REPhase and Phase Difference A pair of sine curves with the same period is given. (a) Find the phase of each curve. (b) Find the phase difference between the curves. (c) Determine whether the curves are in phase or out of phase. (d) Sketch both curves on the same axes. 56. y1=50sin(10t2);y2=50sin10(t20)57RE58RE59REEven and Odd Functions A function is given. (a) Use a graphing device to graph the function. (b) Determine from the graph whether the function is periodic and, if so, determine the period. (c) Determine from the graph whether the function is odd, even, or neither. 60. y = 1 + 2cos x61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RESimple Harmonic Motion A point P moving in simple harmonic motion completes 8 cycles every second. If the amplitude of the motion is 50 cm, find an equation that describes the motion of P as a function of time. Assume that the point P is at its maximum displacement when t = 0.73REDamped Harmonic Motion The top floor of a building undergoes damped harmonic motion after a sudden brief earthquake. At time t = 0 the displacement is at a maximum, 16 cm from the normal position. The damping constant is c = 0.72, and the building vibrates at 1.4 cycles per second. (a) Find a function of the form y = kect cos t to model the motion. (b) Graph the function you found in part (a). (c) What is the displacement at time t = 10 s?1TThe point P in the figure at the left has y-coordinate 45. Find: (a) sin t (b) cos t (c) tan t (d) sec t3TExpress tan t in terms of sin t, if the terminal point determined by t is in Quadrant II.If cost=817 and if the terminal point determined by t is in Quadrant III, find tan t cot t + csc t.6T7T8T9T10TThe graph shown at left is one period of a function of the form y = a sin k(x b). Determine the function.The sine curves y1=30sin(6t2) and y2=30sin(6t3) have the same period. (a) Find the phase of each curve. (b) Find the phase difference between y1 and y2. (c) Determine whether the curves are in phase or out of phase. (d) Sketch both curves on the same axes.13TA mass suspended from a spring oscillates in simple harmonic motion. The mass completes 2 cycles every second, and the distance between the highest point and the lowest point of the oscillation is 10 cm. Find an equation of the form y = a sin t that gives the distance of the mass from its rest position as a function of time.An object is moving up and down in damped harmonic motion. Its displacement at time t = 0 is 16 in.: this is its maximum displacement. The damping constant is c = 0.1, and the frequency is 12 Hz. (a) Find a function that models this motion. (b) Graph the function.(a) The radian measure of an angle is the length of the _____ that subtends the angle in a circle of radius _____. (b) To convert degrees to radians, we multiply by _____. (c) To convert radians to degrees, we multiply by _____.A central angle is drawn in a circle of radius r, as in the figure below. (a) The length of the arc subtended by is s = _____. (b) The area of the sector with central angle is A = _____.Suppose a point moves along a circle with radius r as shown in the figure below. The point travels a distance s along the circle in time t. (a) The angular speed of the point is =. (b) The linear speed of the point is v=. (c) The linear speed v and the angular speed are related by the equation v = _____.Object A is traveling along a circle of radius 2, and Object B is traveling along a circle of radius 5. The objects have the same angular speed. Do the objects have the same linear speed? If not, which object has the greater linear speed?5EFrom Degrees to Radians Find the radian measure of the angle with the given degree measure. Round your answer to three decimal places. 6. 367E8E9E10E11E12E13EFrom Degrees to Radians Find the radian measure of the angle with the given degree measure. Round your answer to three decimal places. 14. 360015E16E17E18E19E20E21E22E23E24E25E26E27E28ECoterminal Angles The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. 29. 50Coterminal Angles The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. 30. 13531E32E33ECoterminal Angles The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. 34. 4535ECoterminal Angles? The measures of two angles in standard position are given. Determine whether the angles are coterminal. 36. 30, 33037ECoterminal Angles? The measures of two angles in standard position are given. Determine whether the angles are coterminal. 38. 323,11339E40E41EFinding a Coterminal Angle Find an angle between 0 and 360 that is coterminal with the given angle. 42. 375Finding a Coterminal Angle Find an angle between 0 and 360 that is coterminal with the given angle. 43. 78044E45E46EFinding a Coterminal Angle Find an angle between 0 and 2 that is coterminal with the given angle. 47. 19648E49EFinding a Coterminal Angle Find an angle between 0 and 2 that is coterminal with the given angle. 50. 1051E52ECircular Arcs Find the length s of the circular arc, the radius r of the circle, or the central angle , as indicated. 53.Circular Arcs Find the length s of the circular arc, the radius r of the circle, or the central angle , as indicated. 54.55E56EFind the length s of the arc that subtends a central angle of measure 3 rad in a circle of radius 5 cm.58EA central angle in a circle of radius 9 m is subtended by an arc of length 14 m. Find the measure of in degrees and radians.An arc of length 15 ft subtends a central angle in a circle of radius 9 ft. Find the measure of in degrees and radians.61EFind the radius r of the circle if an arc of length 20 cm on the circle subtends a central angle of 50.Find the area of the sector shown in each figure. (a) (b)64EFind the area of a sector with central angle 2/3 rad in a circle of radius 10 m.A sector of a circle has a central angle of 145. Find the area of the sector if the radius of the circle is 6 ft.The area of a sector of a circle with a central angle of 140 is 70 m2. Find the radius of the circle.The area of a sector of a circle with a central angle of 5/12 rad is 20 m2. Find the radius of the circle.A sector of a circle of radius 80 mi has an area of 1600 mi2. Find the central angle (in radians) of the sector.The area of a circle is 600 m2. Find the area of a sector of this circle that subtends a central angle of 3 rad.Area of a Sector of a Circle Three circles with radii 1, 2, and 3 ft are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 1 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.Comparing a Triangle and a Sector of a Circle Two wood sticks and a metal rod, each of length 1, are connected to form a triangle with angle 1 at the point P, as shown in the first figure below. The rod is then bent to form an arc of a circle with center P, resulting in a smaller angle 2 at the point P, as shown in the second figure. Find 1, 2, and 1 2.Clocks and Angles In 1 h the minute hand on a clock moves through a complete circle, and the hour hand moves through 112 of a circle. 73. Through how many radians do the minute hand and the hour hand move between 1:00 p.m. and 1:45 p.m. (on the same day)?Clocks and Angles In 1 h the minute hand on a clock moves through a complete circle, and the hour hand moves through 112 of a circle. 74. Through how many radians do the minute hand and the hour hand move between 1:00 p.m. and 6:45 p.m. (on the same day)?Travel Distance A cars wheels are 28 in. in diameter. How far (in mi.) will the car travel if its wheels revolve 10,000 times without slipping?Wheel Revolutions How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?Latitudes Pittsburgh, Pennsylvania, and Miami, Florida, lie approximately on the same meridian. Pittsburgh has a latitude of 40.5N, and Miami has a latitude of 25.5N. Find the distance between these two cities. (The radius of the earth is 3960 mi.)Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has a latitude of 35N, and New Orleans has a latitude of 30N. Find the distance between these two cities. (The radius of the earth is 3960 mi.)Orbit of the Earth Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. [Note: The path of the earth around the sun is actually an ellipse with the sun at one focus (see Section 11.2). This ellipse, however, has very small eccentricity, so it is nearly circular.]Circumference of the Earth The Greek mathematician Eratosthenes (ca. 276195 b.c.) measured the circumference of the earth from the following observations. He noticed that on a certain day the sun shone directly down a deep well in Syene (modern Aswan). At the same time in Alexandria, 500 miles north (on the same meridian), the rays of the sun shone at an angle of 7.2 to the zenith. Use this information and the figure to find the radius and circumference of the earth.81EIrrigation An irrigation system uses a straight sprinkler pipe 300 ft long that pivots around a central point as shown. Because of an obstacle the pipe is allowed to pivot through 280 only. Find the area irrigated by this system.Windshield Wipers The top and bottom ends of a windshield wiper blade are 34 in. and 14 in., respectively, from the pivot point. While in operation, the wiper sweeps through 135. Find the area swept by the blade.The Tethered Cow A cow is tethered by a 100-ft rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze.Fan A ceiling fan with 16-in. blades rotates at 45 rpm. (a) Find the angular speed of the fan in rad/min. (b) Find the linear speed of the tips of the blades in in./min.Radial Saw A radial saw has a blade with a 6-in. radius. Suppose that the blade spins at 1000 rpm. (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the sawteeth in ft/s.Winch A winch of radius 2 ft is used to lift heavy loads. If the winch makes 8 revolutions every 15 s, find the speed at which the load is rising.Speed of a Car The wheels of a car have radius 11 in. and are rotating at 600 rpm. Find the speed of the car in mi/h.Speed at the Equator The earth rotates about its axis once every 23 h 56 min 4 s, and the radius of the earth is 3960 mi. Find the linear speed of a point on the equator in mi/h.Truck Wheels A truck with 48-in.-diameter wheels is traveling at 50 mi/h. (a) Find the angular speed of the wheels in rad/min. (b) How many revolutions per minute do the wheels make?Speed of a Current To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius 0.20 m and rotates at 100 rpm, find the speed of the current in m/s.Bicycle Wheel The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 40 rpm. (a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)Conical Cup A conical cup is made from a circular piece of paper with radius 6 cm by cutting out a sector and joining the edges as shown below. Suppose = 5/3. (a) Find the circumference C of the opening of the cup. (b) Find the radius r of the opening of the cup. [Hint: Use C = 2r.] (c) Find the height h of the cup. [Hint: Use the Pythagorean Theorem.] (d) Find the volume of the cup.Conical Cup In this exercise we find the volume of the conical cup in Exercise 93 for any angle . (a) Follow the steps in Exercise 93 to show that the volume of the cup as a function of is V()=922422,02 (b) Graph the function V. (c) For what angle is the volume of the cup a maximum?WRITE: Different Ways of Measuring Angles The custom of measuring angles using degrees, with 360 in a circle, dates back to the ancient Babylonians, who used a number system based on groups of 60. Another system of measuring angles divides the circle into 400 units, called grads. In this system a right angle is 100 grad, so this fits in with our base 10 number system. Write a short essay comparing the advantages and disadvantages of these two systems and the radian system of measuring angles. Which system do you prefer? Why?A right triangle with an angle is shown in the figure. (a) Label the opposite and adjacent sides of and the hypotenuse of the triangle. (b) The trigonometric functions of the angle are defined as follows: sin=cos=tan= (c) The trigonometric ratios do not depend on the size of the triangle. This is because all right triangles with the same acute angle are __________.The reciprocal identities state that csc=1sec=1cot=1Trigonometric Ratios Find the exact values of the six trigonometric ratios of the angle in the triangle. 3.4ETrigonometric Ratios Find the exact values of the six trigonometric ratios of the angle in the triangle. 5.6E7ETrigonometric Ratios Find the exact values of the six trigonometric ratios of the angle in the triangle. 8.Trigonometric Ratios Find (a) sin and cos , (b) tan and cot , and (c) sec and csc . 9.Trigonometric Ratios Find (a) sin and cos , (b) tan and cot , and (c) sec and csc . 10.Using a Calculator Use a calculator to evaluate the expression. Round your answer to five decimal places. 11. (a) sin 22 (b) cot 2312E13EUsing a Calculator Use a calculator to evaluate the expression. Round your answer to five decimal places. 14. (a) csc 10 (b) sin 4615E16EFinding an Unknown Side Find the side labeled x. In Exercises 17 and 18 state your answer rounded to five decimal places. 17.Finding an Unknown Side Find the side labeled x. In Exercises 17 and 18 state your answer rounded to five decimal places. 18.