4-1 Right Triangle Trigonometry

Transcription

1 Find the exact values of the six trigonometric functions of θ 1 The length of the side opposite θ is 8 is 18, the length of the side adjacent to θ is 14, and the length of the hypotenuse 2 The length of the side opposite θ is 2 hypotenuse is 15 3, the length of the side adjacent to θ is 13, and the length of the Page 1

2 3 The length of the side opposite θ is 9, the length of the side adjacent to θ is 4, and the length of the hypotenuse is 4 The length of the side opposite θ is 12, the length of the side adjacent to θ is 35, and the length of the hypotenuse is 37 5 esolutions Manual - Powered by Cognero The length of the side opposite hypotenuse is 29 θ is, the length of the side adjacent to θ is 26, and the length of the Page 2

3 5 The length of the side opposite θ is hypotenuse is 29, the length of the side adjacent to θ is 26, and the length of the 6 The length of the side opposite θ is 30, the length of the side adjacent to θ is 5 hypotenuse is 35 esolutions 7 Manual - Powered by Cognero, and the length of the Page 3

4 7 The length of the side opposite θ is 6 and the length of the hypotenuse is 10 By the Pythagorean Theorem, the length of the side adjacent to θis = or8 8 The length of the side opposite θ is 8 and the length of the side adjacent to θ is 32 By the Pythagorean Theorem, the length of the hypotenuse is = or8 Use the given trigonometric function value of the acute angle θto find the exact values of the five remaining trigonometric function values of θ esolutions 9sinManual θ= - Powered by Cognero Page 4

5 Use the given trigonometric function value of the acute angle θto find the exact values of the five remaining trigonometric function values of θ 9sin θ= Draw a right triangle and label one acute angle θ Because sin θ= =, label the opposite side 4 and the hypotenuse 5 By the Pythagorean Theorem, the length of the side adjacent to θ is or3 10cos θ= Draw a right triangle and label one acute angle θ Because cos θ= =, label the adjacent side 6 and the hypotenuse 7 By the Pythagorean Theorem, the length of the side opposite θ is or Page 5

6 10cos θ= Draw a right triangle and label one acute angle θ Because cos θ= =, label the adjacent side 6 and the hypotenuse 7 By the Pythagorean Theorem, the length of the side opposite θ is or 11tan θ=3 Draw a right triangle and label one acute angle θ Because tan θ=, label the side opposite θ 3 and =3 or the adjacent side 1 By the Pythagorean Theorem, the length of the hypotenuse is or Page 6

7 11tan θ=3 Draw a right triangle and label one acute angle θ Because tan θ=, label the side opposite θ 3 and =3 or the adjacent side 1 By the Pythagorean Theorem, the length of the hypotenuse is or 12sec θ=8 Draw a right triangle and label one acute angle θ Because sec θ= =8or, label the hypotenuse 8 and the adjacent side 1 By the Pythagorean Theorem, the length of the side adjacent to θ is = or3 Page 7

8 12sec θ=8 Draw a right triangle and label one acute angle θ Because sec θ= =8or, label the hypotenuse 8 and the adjacent side 1 By the Pythagorean Theorem, the length of the side adjacent to θ is = or3 13cos θ= Draw a right triangle and label one acute angle θ Because cos θ= =, label the adjacent side 5 and the hypotenuse 9 By the Pythagorean Theorem, the length of the side opposite θ is or2 Page 8

9 13cos θ= Draw a right triangle and label one acute angle θ Because cos θ= =, label the adjacent side 5 and the hypotenuse 9 By the Pythagorean Theorem, the length of the side opposite θ is or2 14tan θ= Draw a right triangle and label one acute angle θ Because tan θ= =, label the side opposite θ 5 and adjacent side 4 By the Pythagorean Theorem, the length of the hypotenuse is or Page 9

10 14tan θ= Draw a right triangle and label one acute angle θ Because tan θ= =, label the side opposite θ 5 and adjacent side 4 By the Pythagorean Theorem, the length of the hypotenuse is or 15cot θ=5 Draw a right triangle and label one acute angle θ Because cot θ=,labelthesideadjacentto θ as =5 or 5 and the opposite side 1 By the Pythagorean Theorem, the length of the hypotenuse is or 16csc θ=6 esolutions Manual - Powered by Cognero Draw a right triangle and label one acute angle θ Because csc θ = Page 10 =6 or, label the hypotenuse 6 and the

11 16csc θ=6 Draw a right triangle and label one acute angle θ Because csc θ = =6 or, label the hypotenuse 6 and the side opposite θ as 1 By the Pythagorean Theorem, the length of the adjacent side is or 17sec θ= Draw a right triangle and label one acute angle θ Because sec θ= =, label the hypotenuse 9 and the side opposite θ as 2 By the Pythagorean Theorem, the length of the opposite side is or Page 11

12 17sec θ= Draw a right triangle and label one acute angle θ Because sec θ= =, label the hypotenuse 9 and the side opposite θ as 2 By the Pythagorean Theorem, the length of the opposite side is or 18sin θ= Draw a right triangle and label one acute angle θ Because sin θ= =, label the side opposite θ as 8 and the hypotenuse 13 By the Pythagorean Theorem, the length of the adjacent side is or Page 12

13 18sin θ= Draw a right triangle and label one acute angle θ Because sin θ= =, label the side opposite θ as 8 and the hypotenuse 13 By the Pythagorean Theorem, the length of the adjacent side is or Find the value of x Round to the nearest tenth 19 An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite θ 20 An acute angle measure and the length of the side opposite θ are given, so the tangent function can be used to find themanual length- Powered of the side adjacent to θ esolutions by Cognero Page 13

14 20 An acute angle measure and the length of the side opposite θ are given, so the tangent function can be used to find the length of the side adjacent to θ 21 An acute angle measure and the length of the hypotenuse are given, so the cosine function can be used to find the length of the side adjacent to θ 22 An acute angle measure and the length of the side adjacent to θ are given, so the cosine function can be used to find the length of the hypotenuse 23 Manual - Powered by Cognero esolutions Page 14

15 23 An acute angle measure and the length of the side opposite θ are given, so the sine function can be used to find the length of the hypotenuse 24 An acute angle measure and the length of the side adjacent to θ are given, so the tangent function can be used to find the length of the side opposite θ 25 An acute angle measure and the length of the side opposite θ are given, so the tangent function can be used to find the length of the side adjacent to θ esolutions 26 Manual - Powered by Cognero Page 15

16 26 An acute angle measure and the length of the side opposite θ are given, so the sine function can be used to find the length of the hypotenuse 27MOUNTAIN CLIMBINGAteamofclimbersmustdeterminethewidthofaravineinordertosetup equipment to cross it If the climbers walk 25 feet along the ravine from their crossing point, and sight the crossing point on the far side of the ravine to be at a 35º angle, how wide is the ravine? An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the length of the opposite side Therefore, the ravine is about 175 feet wide 28SNOWBOARDINGBradbuiltasnowboardingrampwithaheightof35feetandan18º incline a Draw a diagram to represent the situation b Determine the length of the ramp a Draw a diagram of a right triangle and label one acute angle 18º and the opposite side 35 feet b Because an acute angle measure and opposite side length are given, the sine function can be used to find the Page 16 length of the hypotenuse

17 Therefore, the ravine is about 175 feet wide 28SNOWBOARDINGBradbuiltasnowboardingrampwithaheightof35feetandan18º incline a Draw a diagram to represent the situation b Determine the length of the ramp a Draw a diagram of a right triangle and label one acute angle 18º and the opposite side 35 feet b Because an acute angle measure and opposite side length are given, the sine function can be used to find the length of the hypotenuse Therefore, the length of the ramp is about 113 feet 29DETOURTrafficisdetouredfromElwoodAve,left08mileonMapleSt,andthenrightonOakSt,which intersectselwoodaveata32 angle a Draw a diagram to represent the situation b Determine the length of Elwood Ave that is detoured a Draw a right triangle with acute angle 32º and opposite side length 08 mile Label the side opposite the 32º angle Maple St, the hypotenuse Oak St, and the adjacent side Elwood Ave b Because an acute angle and the length of opposite side are given, the tangent function can be used to find the adjacent side length esolutions Manual - Powered by Cognero Therefore, the length of Elwood Ave that is detoured is about 13 miles Page 17 30PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet,

18 Therefore, the length of the ramp is about 113 feet 29DETOURTrafficisdetouredfromElwoodAve,left08mileonMapleSt,andthenrightonOakSt,which intersectselwoodaveata32 angle a Draw a diagram to represent the situation b Determine the length of Elwood Ave that is detoured a Draw a right triangle with acute angle 32º and opposite side length 08 mile Label the side opposite the 32º angle Maple St, the hypotenuse Oak St, and the adjacent side Elwood Ave b Because an acute angle and the length of opposite side are given, the tangent function can be used to find the adjacent side length Therefore, the length of Elwood Ave that is detoured is about 13 miles 30PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet, causing him to drift at an 8º angle How far from the drop zone will the paratrooper land? Because an acute angle and the length of the side that is adjacent to the angle are given, the tangent function can be used to find the length of the opposite side So,Manual the paratrooper land esolutions - Powered bywill Cognero about 190 feet away from the drop zone Find the measure of angle θ Round to the nearest degree, if necessary Page 18

19 Therefore, the length of Elwood Ave that is detoured is about 13 miles 30PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet, causing him to drift at an 8º angle How far from the drop zone will the paratrooper land? Because an acute angle and the length of the side that is adjacent to the angle are given, the tangent function can be used to find the length of the opposite side So, the paratrooper will land about 190 feet away from the drop zone Find the measure of angle θ Round to the nearest degree, if necessary 31 Because the lengths of the sides opposite and adjacent to θ are given, the tangent function can be used to find θ 32 Because the length of the hypotenuse and side adjacent to θare given, the cosine function can be used to find θ Page 19

20 32 Because the length of the hypotenuse and side adjacent to θare given, the cosine function can be used to find θ 33 Because the length of the hypotenuse and side opposite θare given, the sine function can be used to find θ 34 Because the lengths of the sides opposite and adjacent to θare given, the tangent function can be used to find θ 35 Because the lengths of the sides opposite and adjacent to θare given, the tangent function can be used to find θ Page 20

21 35 Because the lengths of the sides opposite and adjacent to θare given, the tangent function can be used to find θ 36 Because the length of the hypotenuse and side adjacent to θare given, the cosine function can be used to find θ 37 Because the length of the hypotenuse and side opposite θare given, the sine function can be used to find θ 38 Because the lengths of the sides opposite and adjacent to θare given, the tangent function can be used to find θ Page 21

22 38 Because the lengths of the sides opposite and adjacent to θare given, the tangent function can be used to find θ 39PARASAILING KayladecidedtotryparasailingShewasstrappedintoaparachutetowedbyaboatAn800foot line connected her parachute to the boat, which was at a 32º angle of depression below her How high above the water was Kayla? The angle of elevation from the boat to the parachute is equivalent to the angle of depression from the parachute to the boat because the two angles are alternate interior angles, as shown below Because an acute angle and the hypotenuse are given, the sine function can be used to find x Therefore, Kayla was about 424 feet above the water 40OBSERVATION WHEEL TheLondonEyeisa135-meter-tall observation wheel If a passenger at the top of the wheel sights the London Aquarium at a 58º angle of depression, what is the distance between the aquarium and the London Eye? The angle of depression from the top of the wheel to the aquarium is 58º, which means that the angle of elevation from the aquarium to the top of the wheel is also 58º because they are alternate interior angles Draw a diagram of Page 22 a right triangle and label one acute angle 58º and the opposite side 135 m

23 Therefore, Kayla was about 424 feet above the water 40OBSERVATION WHEEL TheLondonEyeisa135-meter-tall observation wheel If a passenger at the top of the wheel sights the London Aquarium at a 58º angle of depression, what is the distance between the aquarium and the London Eye? The angle of depression from the top of the wheel to the aquarium is 58º, which means that the angle of elevation from the aquarium to the top of the wheel is also 58º because they are alternate interior angles Draw a diagram of a right triangle and label one acute angle 58º and the opposite side 135 m Because an acute angle and the side length opposite the angle are given, the tangent function can be used to find x Therefore, the distance between the aquarium and the London Eye is about 84 meters 41ROLLER COASTER Onarollercoaster,375feetoftrackascendata55º angle of elevation to the top before the first and highest drop a Draw a diagram to represent the situation b Determine the height of the roller coaster a Draw a diagram of a right triangle and label one acute angle 55º and the hypotenuse 375 feet b Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find the length of the opposite side Page 23

24 Therefore, the distance between the aquarium and the London Eye is about 84 meters 41ROLLER COASTER Onarollercoaster,375feetoftrackascendata55º angle of elevation to the top before the first and highest drop a Draw a diagram to represent the situation b Determine the height of the roller coaster a Draw a diagram of a right triangle and label one acute angle 55º and the hypotenuse 375 feet b Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find the length of the opposite side Therefore, the height of the roller coaster is about 307 feet 42SKI LIFTAcompanyisinstallinganewskiliftona225-meter-high mountain that will ascend at a 48º angle of elevation a Draw a diagram to represent the situation b Determine the length of cable the lift requires to extend from the base to the peak of the mountain a Draw a diagram of a right triangle and label one acute angle 48º and the opposite side 225 meters b Because an acute angle and the length of the side opposite the angle are given, you can use the sine function to find the length of the hypotenuse So, the company will need about 303 meters of cable esolutions Manual - Powered BothDerekandSamare5feet10inchestallDereklooksata10-foot by Cognero 43BASKETBALL basketball goal withpage an 24 angleofelevationof29,andsamlooksatthegoalwithanangleofelevationof43 IfSamisdirectlyinfrontof Derek, how far apart are the boys standing?

25 So, the company will need about 303 meters of cable 43BASKETBALLBothDerekandSamare5feet10inchestallDereklooksata10-foot basketball goal with an angleofelevationof29,andsamlooksatthegoalwithanangleofelevationof43 IfSamisdirectlyinfrontof Derek, how far apart are the boys standing? Draw a diagram to model the situation The vertical distance from the boys' heads to the rim is 10(12) [5(12) + 10] or 50 inches Label the horizontal distance between Sam and Derek as x and the horizontal distance between Sam and the goal as y From the smaller right triangle, you can use the tangent function to find y From the larger right triangle, you can use the tangent function to find x Therefore, Derek and Sam are standing about 366 inches or 31 feet apart 44PARISAtouristonthefirstobservationleveloftheEiffelTowersightstheMuséeD Orsay at a 14º angle of depressionatouristonthethirdobservationlevel,located219metersdirectlyabovethefirst,sightsthemusée D Orsay at a 68º angle of depression a Draw a diagram to represent the situation b DeterminethedistancebetweentheEiffelTowerandtheMuséeD Orsay Page 25

26 Therefore, Derek and Sam are standing about 366 inches or 31 feet apart 44PARISAtouristonthefirstobservationleveloftheEiffelTowersightstheMuséeD Orsay at a 14º angle of depressionatouristonthethirdobservationlevel,located219metersdirectlyabovethefirst,sightsthemusée D Orsay at a 68º angle of depression a Draw a diagram to represent the situation b DeterminethedistancebetweentheEiffelTowerandtheMuséeD Orsay a st rd b Because the angles of depression from the 1 and 3 levelstothemuséed Orsay are 14 and 68, st rd respectively,theanglesofelevationfromthemuséed Orsaytothe1 and 3 levels are also 14 and 68, respectively Use the tangent function to write an equation for the smaller right triangle in terms of y Next, use the tangent function to write an equation for the larger right triangle in terms of y Set the equations that you found for each triangle equal to one another and solve for x Therefore,thedistancebetweentheEiffelTowerandtheMuséeD Orsay is about 2310 meters 45LIGHTHOUSETwoshipsarespottedfromthetopofa156-foot lighthouse The first ship is at a 27º angle of depression, and the second ship is directly behind the first at a 7º angle of depression a Draw a diagram to represent the situation b Determine the distance between the two ships a Page 26

27 Therefore,thedistancebetweentheEiffelTowerandtheMuséeD Orsay is about 2310 meters 45LIGHTHOUSETwoshipsarespottedfromthetopofa156-foot lighthouse The first ship is at a 27º angle of depression, and the second ship is directly behind the first at a 7º angle of depression a Draw a diagram to represent the situation b Determine the distance between the two ships a b From the smaller right triangle, you can use the tangent function to find y From the larger right triangle, you can use the tangent fuction to find x Therefore, the distance between the two ships is about 964 feet 46MOUNT RUSHMOREThefacesofthepresidentsatMountRushmoreare60feettallAvisitorseesthetop of George Washington s head at a 48º angle of elevation and his chin at a 4476º angle of elevation Find the height of Mount Rushmore From the smaller triangle, you can use the tangent function to find y Page 27

28 Therefore, the distance between the two ships is about 964 feet 46MOUNT RUSHMOREThefacesofthepresidentsatMountRushmoreare60feettallAvisitorseesthetop of George Washington s head at a 48º angle of elevation and his chin at a 4476º angle of elevation Find the height of Mount Rushmore From the smaller triangle, you can use the tangent function to find y From the larger triangle, you can use the tangent function to find y, too Next, set the two equations equal to one another to solve for x Therefore, Mount Rushmore is about 500 feet tall Solve each triangle Round side measures to the nearest tenth and angle measures to the nearest degree 47 Use trigonometric functions to find b and c Page 28

29 Therefore, Mount Rushmore is about 500 feet tall Solve each triangle Round side measures to the nearest tenth and angle measures to the nearest degree 47 Use trigonometric functions to find b and c Because the measures of two angles are given, B can be found by subtracting A from Therefore, b 165,c Use trigonometric functions to find y and z Because the measures of two angles are given, X can be found by subtracting Z from Therefore, y 371,z Use the Pythagorean Theorem to find r Page 29

30 Because the measures of two angles are given, X can be found by subtracting Z from 4-1 Right Triangle Trigonometry Therefore, y 371,z Use the Pythagorean Theorem to find r Use the tangent function to find P Because the measures of two angles are now known, you can find Q by subtracting P from Therefore, P 43, Q 47, and r Use the Pythagorean Theorem to find d Use the cosine function to find D Because the measures of two angles are now known, you can find E by subtracting D from Therefore, D 77, E 13, and d Use trigonometric functions to find j and k Page 30

31 Because the measures of two angles are now known, you can find E by subtracting D from 4-1 Right Triangle Trigonometry Therefore, D 77, E 13, and d Use trigonometric functions to find j and k Because the measures of two angles are given, K can be found by subtracting J from Therefore, j 180,k Use the Pythagorean Theorem to find y Use the sine function to find W Because the measures of two angles are now known, you can find Y by subtracting W from Therefore, and y Use trigonometric functions to find f and h Page 31

32 Because the measures of two angles are now known, you can find Y by subtracting W from 4-1 Right Triangle Trigonometry Therefore, and y Use trigonometric functions to find f and h Because the measures of two angles are given, H can be found by subtracting F from Therefore, f 196,h Use the Pythagorean Theorem to find t Use the tangent function to find R Because the measures of two angles are now known, you can find S by subtracting R from Therefore, and t 81 55BASEBALL Michael s seat at a game is 65 feet behind home plate His line of vision is 10 feet above the field a Draw a diagram to represent the situation b What is the angle of depression to home plate? a Page 32 b Michael s angle of depression to home plate is equivalent to the angle of elevation from home plate to where he is sitting

33 Because the measures of two angles are now known, you can find S by subtracting R from 4-1 Right Triangle Trigonometry Therefore, and t 81 55BASEBALL Michael s seat at a game is 65 feet behind home plate His line of vision is 10 feet above the field a Draw a diagram to represent the situation b What is the angle of depression to home plate? a b Michael s angle of depression to home plate is equivalent to the angle of elevation from home plate to where he is sitting Use the tangent function to find θ Therefore, the angle of depression to home plate is about 56HIKING Jessica is standing 2 miles from the center of the base of Pikes Peak, and looking at the summit of the mountain, which is 14 miles from the base a Draw a diagram to represent the situation b With what angle of elevation is Jessica looking at the summit of the mountain? a b Use the tangent function to find the angle of elevation Therefore, the angle of elevation is about Find the exact value of each expression without using a calculator 57sin 60 Draw a diagram of a 30º-60º-90º triangle Page 33

34 Therefore, the angle of elevation is about Find the exact value of each expression without using a calculator 57sin 60 Draw a diagram of a 30º-60º-90º triangle The length of the side opposite the 60 angle is x and the length of the hypotenuse is 2x 58cot30 Draw a diagram of a 30º-60º-90º triangle The length of the side adjacent to the 30º angle is x and the length of the opposite side is x 59sec30 Draw a diagram of a 30º-60º-90º triangle The length of the hypotenuse is 2x and the length of the side adjacent to the 30º angle is x Page 34

35 59sec30 Draw a diagram of a 30º-60º-90º triangle The length of the hypotenuse is 2x and the length of the side adjacent to the 30º angle is x 60cos45 Draw a diagram of a 45º-45º-90º triangle The side length is x and the hypotenuse is x 61tan60 Draw a diagram of a 30º-60º-90º triangle esolutions Manual - Powered by Cognero The length of the side opposite the 60º is x and the length of the adjacent side is x Page 35

36 61tan60 Draw a diagram of a 30º-60º-90º triangle The length of the side opposite the 60º is x and the length of the adjacent side is x 62csc45 Draw a diagram of a 45º-45º-90º triangle The length of the hypotenuse is x and the side length is x Without using a calculator, find the measure of the acute angle θ that satisfies the given equation 63tan θ = 1 Because tan θ = 1 and tan θ=, it follows that =1Inthe opposite an acute angle is 1 and the adjacent side length is also 1 So, esolutions Manual - Powered by Cognero Therefore, θ = 45 triangle below, the side length =1 Page 36

37 Without using a calculator, find the measure of the acute angle θ that satisfies the given equation 63tan θ = 1 Because tan θ = 1 and tan θ=, it follows that =1Inthe triangle below, the side length =1 opposite an acute angle is 1 and the adjacent side length is also 1 So, Therefore, θ = 45 64cos θ= Because cos θ= and cos θ= length adjacent to the 30º angle is, it follows that = and the hypotenuse is 2 So, In the = triangle below, the side Therefore, θ = 30 65cot θ = Because cot θ = and cot θ=, it follows that = In the length that is adjacent to the 60º angle is 1 and the length of the opposite side is esolutions Manual - Powered by Cognero Therefore, θ = 60 triangle below, the side So, = or Page 37

38 Therefore, θ = 30 65cot θ = Because cot θ = and cot θ=, it follows that = In the triangle below, the side length that is adjacent to the 60º angle is 1 and the length of the opposite side is = So, or Therefore, θ = 60 66sin θ= Because sin θ= and sin θ=, it follows that length opposite an acute angle is 1 and the hypotenuse is = In the So, = triangle below, the side or Therefore, θ =45 67csc θ = 2 Because csc θ = 2 and csc θ=, it follows that =2or In the hypotenuse is 2 and the side length that is opposite the 30º angle is 1 So, triangle below, the = or2 Therefore, θ =30 68sec θ = 2 Page 38

39 Therefore, θ =45 67csc θ = 2 Because csc θ = 2 and csc θ=, it follows that =2or In the hypotenuse is 2 and the side length that is opposite the 30º angle is 1 So, triangle below, the = or2 Therefore, θ =30 68sec θ = 2 Because sec θ = 2 and sec θ=, it follows that =2or In the hypotenuse is 2 and the side length that is adjacent to the 60º angle is 1 So, triangle below, the = or2 Therefore, θ = 60 Without using a calculator, determine the value of x 69 Because an acute angle, the hypotenuse, and the length of the side adjacent to the angle are given, the cosine function can be used to find x Using the properties of triangles, you can find that cos 30º = Page 39

40 Therefore, θ = 60 Without using a calculator, determine the value of x 69 Because an acute angle, the hypotenuse, and the length of the side adjacent to the angle are given, the cosine function can be used to find x Using the properties of triangles, you can find that cos 30º = 70 Because the triangle is a triangle, the legs are the same length 71SCUBA DIVINGAscubadiverlocated20feetbelowthesurfaceofthewaterspotsashipwreckata70º angle of depression After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57º angle of depression Draw a diagram to represent the situation, and determine the depth of the shipwreck First, draw a diagram to represent the situation Page 40

41 71SCUBA DIVINGAscubadiverlocated20feetbelowthesurfaceofthewaterspotsashipwreckata70º angle of depression After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57º angle of depression Draw a diagram to represent the situation, and determine the depth of the shipwreck First, draw a diagram to represent the situation Label the horizontal distance from the shipwreck to the point on the ocean floor below the diver as x Label the vertical distance from the point 20 feet below sea level to the point 45 feet below sea level as y Find the complementary angle for each angle of depression Use the tangent function to write an equation for the smaller right triangle in terms of x Use the tangent function to write an equation for the larger right triangle in terms of x Set the two equations equal to one another to solve for y Therefore, the depth of the shipwreck is or about 100 feet Find the value of cos θif θ is the measure of the smallest angle in each type of right triangle Draw a diagram of a triangle The smallest angle will be the angle opposite the side with a length of 3 Page 41

42 Therefore, the depth of the shipwreck is or about 100 feet Find the value of cos θif θ is the measure of the smallest angle in each type of right triangle Draw a diagram of a triangle The smallest angle will be the angle opposite the side with a length of Draw a diagram of a triangle The smallest angle will be the angle opposite the side with a length of 5 74SOLAR POWERFindthetotalareaofthepanelshownbelow The area of the panel is given by, where the width is 10 feet and the length is unknown Notice that the length of the panel is also the hypotenuse of a right triangle with an acute angle of 55º and an opposite side length of 35 feet The sine function can be used to find the hypotenuse of the triangle So, the length of the panel is 427 feet Find the area Page 42

43 74SOLAR POWERFindthetotalareaofthepanelshownbelow The area of the panel is given by, where the width is 10 feet and the length is unknown Notice that the length of the panel is also the hypotenuse of a right triangle with an acute angle of 55º and an opposite side length of 35 feet The sine function can be used to find the hypotenuse of the triangle So, the length of the panel is 427 feet Find the area Therefore, the area of the panel is 427 square feet Without using a calculator, insert the appropriate symbol >, <, or = to complete each equation 75sin45 cot60 Use a graphing calculator to find sin 45º and cot 60º To find cot 60º, find sin 45º 0707 cot 60º 0577 Therefore, sin 45º > cot 60º 76tan60 cot30 Use a graphing calculator to find tan 60º and cot 30º To find cot 30º, find tan 60º 1732 cot 30º 1732 Therefore,tan60 =cot30 77cos30 csc45 Use a graphing calculator to find cos 30º and csc 45º To find csc 45º, find cos 30º 0866 csc 45º 1414 Therefore,cos30 <csc45 78cos30 sin60 Page 43

44 Use a graphing calculator to find tan 60º and cot 30º To find cot 30º, find 4-1 tan 60º 1732 cot 30º Triangle 1732 Right Trigonometry Therefore,tan60 =cot30 77cos30 csc45 Use a graphing calculator to find cos 30º and csc 45º To find csc 45º, find cos 30º 0866 csc 45º 1414 Therefore,cos30 <csc45 78cos30 sin60 Use a graphing calculator to find cos 30º and sin 60º cos 30º 0866 sin 60º 0866 Therefore,cos30 =sin60 79sec45 csc60 sec45 csc60 Use a graphing calculator to find sec 45º and csc 60º To find sec 45ºfind, and to find csc 60ºfind sec 45º 1414 csc 60º 1155 Therefore, sec 45º > csc60 80tan45 sec30 tan45 sec30 Use a graphing calculator to find tan 45º and sec 30º To find sec 30º, find tan 45º = 1 sec 30º 1155 Therefore,tan45 <sec30 81ENGINEERINGDeterminethedepthoftheshaftatthelargeendd of the air duct shown below if the taper of the duct is 35º Draw a diagram of the side-view of the air duct Label the lengths of the top and bottom of the duct 48 in because 4 feet = 4 12 or 48 inches Page 44

45 Use a graphing calculator to find tan 45º and sec 30º To find sec 30º, find 4-1 tan 45º = 1 sec 30ºTriangle 1155 Right Trigonometry Therefore,tan45 <sec30 81ENGINEERINGDeterminethedepthoftheshaftatthelargeendd of the air duct shown below if the taper of the duct is 35º Draw a diagram of the side-view of the air duct Label the lengths of the top and bottom of the duct 48 in because 4 feet = 4 12 or 48 inches Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find x Find the length of the large end of the duct Therefore, the length of the large end of the duct is about 179 inches 82MULTIPLEREPRESENTATIONSIn this problem, you will investigate trigonometric functions of acute angles and their relationship to points on the coordinate plane a GRAPHICALLetP(x, y) be a point in the first quadrant Graph the line through point P and the origin Form a right triangle by connecting the points P, (x, 0), and the origin Label the lengths of the legs of the triangle in terms of x or y Label the length of the hypotenuse as r and the angle the line makes with the x-axis θ b ANALYTICAL Express the value of r in terms of x and y c ANALYTICAL Express sin θ, cos θ, and tan θ in terms of x, y, and/or r d VERBAL Under what condition can the coordinates of point P be expressed as (cos θ, sin θ)? e ANALYTICALWhichtrigonometricratioinvolving θ corresponds to the slope of the line? f ANALYTICALFindanexpressionfortheslopeofthelineperpendiculartothelineinparta in terms of θ a Page 45

46 d VERBAL Under what condition can the coordinates of point P be expressed as (cos θ, sin θ)? e ANALYTICALWhichtrigonometricratioinvolving θ corresponds to the slope of the line? f ANALYTICALFindanexpressionfortheslopeofthelineperpendiculartothelineinparta in terms of θ 4-1 Right Triangle Trigonometry a b By the Pythagorean Theorem, x + y = r So, r = c Use the definitions of the sine, cosine, and tangent functions and the diagram from part a to find sin θ, cos θ, and tan θ d Sample answer: Because cos θ = and sin θ =, when r = 1, cos θ= or x and sin θ= or y Therefore,apointP(x, y) can be written as (cos θ, sin θ) when r = 1 e Find the slope of the line using the points (0, 0) and (x, y) From part c, Therefore, tan θ corresponds to the slope of the line ftheslopeofthelineperpendiculartoalinewithslope it follows that is the negative reciprocal or Because, Therefore, an expression for the slope of the line perpendicular to the line in part a is cot θ 83PROOF Prove that if θ is an acute angle of a right triangle, then tan θ = and cot θ = Sample answer: For an acute angle θ of a right triangle, sin θ = Using these definitions, 84ERROR esolutions Manual - ANALYSIS Powered by Cognero Jason = =,cos θ = =tan θ Similarly,, tan θ = = =,and cot θ = =cot θ and Nadina know the value of sin θ = a and are asked to find csc θ Jason sayspage that46 this is not possible, but Nadina disagrees Is either of them correct? Explain your reasoning

47 Using these definitions, = = =tan θ Similarly, = = =cot θ 84ERROR ANALYSIS Jason and Nadina know the value of sin θ = a and are asked to find csc θ Jason says that this is not possible, but Nadina disagrees Is either of them correct? Explain your reasoning Nadina; the cosecant function is the reciprocal function of the sine function Therefore, if sin θ = a, then csc θ = 0, where a 85Writing in Math Explain why the six trigonometric functions are transcendental functions The trigonometric functions are transcendental functions because they cannot be expressed in terms of algebraic operations For example, there is no way to find the value of θin y = cos θby adding, subtracting, multiplying, or dividing a constant and θ or raising θ to a rational power 86CHALLENGEWrite an expression in terms of θ for the area of the scalene triangle shown Explain Sample answer: If you draw the height of the triangle, it forms two right triangles, as shown Use the sine function to find h Substitute the expression for h into the formula for the area of a triangle, A = Therefore, an expression in terms of θ for the area of the triangle is A = bh 87PROOF Prove that if θ is an acute angle of a right triangle, then (sin θ)2 + (cos θ)2 = 1 esolutions Manual - Poweredof byacognero Draw a diagram right triangle with an acute angle θ Page 47

48 87 PROOF Prove that if θ is an acute angle of a right triangle, then (sin θ) 2 + (cos θ) 2 = 1 SOLUTION: Draw a diagram of a right triangle with an acute angle θ From the definitions of the sine and cosine functions, sin θ = and cos θ = By the Pythagorean Theorem, c 2 = a 2 + b 2 = + = = = 1 Therefore, (sin θ) 2 + (cos θ) 2 = 1 REASONING If A and B are the acute angles of a right triangle and m A < m B, determine whether each statement is true or false If false, give a counterexample 88 sin A < sin B SOLUTION: Find sin A and sin B for various angle measures of a right triangle, where m A < m B m A m B sin A sin B Sample answer: From the table, it appears that when m A < m B, sin A < sin B Therefore, the statement is true Page 48

49 89 cos A < cos B SOLUTION: False; sample answer: In ABC, the m B < m A, cos B 07986, and cos A Therefore, cos B > cos A, and thus the statement is false 90 tan A < tan B SOLUTION: Find tan A and tan B for various angle measures of a right triangle, where m A < m B m A m B tan A tan B Sample answer: From the table, it appears that when m A < m B, tan A < tan B Therefore, the statement is true 91 Writing in Math Notice on a graphing calculator that there is no key for finding the secant, cosecant, or cotangent of an angle measure Explain why you think this might be so SOLUTION: Sample answer: Since cosine is the inverse of secant, sine is the inverse of cosecant, and tangent is the inverse of cotangent, you can find the value of the secant, cosecant, or cotangent by finding the value of its inverse and using the reciprocal key on your calculator 92 ECONOMICS The Consumer Price Index (CPI) measures inflation It is based on the average prices of goods and services in the United States, with the annual average for the years set at an index of 100 The table shown gives some annual average CPI values from 1955 to 2005 Find an exponential model relating this data (year, CPI) by linearizing the data Let x = 0 represent 1955 Then use your model to predict the CPI for 2025 SOLUTION: Linearize the data by finding (x, ln y) Page 49

50 x ln y Calculate the linear regression Please note that the regression was calculated using the actual values of ln y as opposed to the rounded values shown in the table above The rounded regression equation is Graph the linearized data Replace with ln y and solve for y To find the CPI for 2025, find y when x = or 70 According to this model, the CPI in 2025 will be about 5232 Solve each equation Round to the nearest hundredth 93 e 5x = 24 SOLUTION: Page 50

51 94 2e x 7 6 = 0 SOLUTION: 2e x 7 6 = 0 Sketch and analyze the graph of each function Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing 95 f (x) = 3 x 2 SOLUTION: Evaluate the function for several x-values in its domain x f (x) Use a smooth curve to connect each of the ordered pairs Page 51

52 96 f (x) = 2 3x SOLUTION: Evaluate the function for several x-values in its domain x f (x) Use a smooth curve to connect each of the ordered pairs Page 52

53 97 f (x) = 4 x + 6 SOLUTION: Evaluate the function for several x-values in its domain x f (x) Use a smooth curve to connect each of the ordered pairs Page 53

54 Solve each equation 98 = SOLUTION: The LCD is (x + 4)(2x 1) 99 = + SOLUTION: The LCD is (x + 1)(x 5) Because the original equation is not defined when x = 5, you can eliminate this extraneous solution Therefore, the only solution is 4 Page 54

55 100 = SOLUTION: = The LCD is (3x + 2)(x + 1) Page 55

56 101 NEWSPAPERS The circulation in thousands of papers of a national newspaper is shown a Let x equal the number of years after 2001 Create a scatter plot of the data b Determine a power function to model the data c Use the function to predict the circulation of the newspaper in 2015 SOLUTION: a Enter the data into a graphing calculator and create a scatter plot b Use the power regression function on the graphing calculator to find values for a and b Rounding to the nearest thousandth, a power function that can be used to model the data is f (x) = x 0149 c Graph the regression equation using a graphing calculator To predict the circulation of the newspaper in 2015, use the TRACE function on the graphing calculator Find the value of y when x = or 14 So, the circulation in 2015 will be about 611,068 Page 56

57 102 SAT/ACT In the figure below, what is the value of z? A 15 B 15 C 15 D 30 E 30 SOLUTION: First, find x Notice that this is a 45º-45º-90º triangle Therefore, the legs of the triangle are the same length Solve for y So, the length of each side is 3(5) or 15 Use the Pythagorean Theorem to find z Therefore, the correct answer is B 103 REVIEW Joseph uses a ladder to reach a window 10 feet above the ground If the ladder is 3 feet away from the wall, how long should the ladder be? F 939 ft G 1044 ft H 1123 ft J 1205 ft SOLUTION: Draw a diagram where x represents the length of the ladder Use the Pythagorean Theorem to find x Therefore, the correct answer is G Page 57

58 104 A person holds one end of a rope that runs through a pulley and has a weight attached to the other end Assume that the weight is at the same height as the person s hand What is the distance from the person's hand to the weight? A 78 ft B 105 ft C 129 ft D 143 ft SOLUTION: Draw a diagram where x represents the distance from the person's hand to the weight Because an acute angle and the length of the side adjacent to the angle are given, you can use the tangent function to find x The distance from the person's hand to the weight is about 78 feet Therefore, the correct answer is A Page 58

59 105 REVIEW A kite is being flown at a 45 angle The string of the kite is 120 feet long How high is the kite above the point at which the string is held? F 60 ft G 60 ft H 60 ft J 120 ft SOLUTION: Draw a diagram where x represents the distance from where the string is being held to the kite Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find x The kite is 60 ft above the point where the string is being held Therefore, the correct answer is G Page 59