Solving Problems with Trigonometry

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William Harper
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1 Cnnectins Have yu ever... Slving Prblems with Trignmetry Mdeled a prblem using a right triangle? Had t find the height f a flagple r clumn? Wndered hw far away a helicpter was? Trignmetry can be used t slve real-wrld situatins invlving triangles. Fr instance, a pint n the grund, the tp f a flagple, and the bttm f a flagple frm a right triangle. Suppse yu knw the height f the flagple and the angle f elevatin frm the bttm t the tp f the flagple. Hw can yu use trignmetry t find the distance frm the pint n the grund t the bttm f the flagple? Trignmetry is the study f the relatinships between the sides and angles f a triangle. It can be used t find an unknwn side r angle f a right triangle. A trignmetric rati is a fractin frmed by tw sides f a triangle. Three imprtant trignmetric ratis are sine (sin), csine (cs) and tangent (tan). T find these ratis in a right triangle, start with ne f the acute angles, such as the ne labeled using the uppercase Greek letter theta (H) in the diagram. Ntice which sides are the hyptenuse, ppsite, and adjacent sides and hw t calculate sine, csine, and tangent. Yu can use these definitins t find unknwn side lengths f a right triangle. sin H = cs H = ppsite hyptenuse adjacent hyptenuse ppsite adjacent H hyptenuse adjacent ppsite 1

2 Learn It! Essential Math Skills Using Trignmetry t Find a Distance Here s an eample f a prblem using trignmetry. In the figure belw, tw side lengths are given, an angle measure is given, and the side labeled is unknwn. What if yu want t find the value f? sin(31) = 7 cs(31) = 6 7 tan(31) = c Math Tip Remember: Every right triangle has tw acute angles and ne right angle. The lngest side f every right triangle is the hyptenuse. The ther tw sides are called legs. Math Tip Yu can use yur calculatr t find the sine, csine, r tangent f an angle. Press the sin, cs, r tan buttn, then enter the angle measurement, and then press enter. Yu can write an equatin t find the value f. Tw f the trignmetric ratis have the length yu want t find, s either ne will d. Let s use sin(31) = 7. T find the value f, yu can multiply bth sides by 7. sin(31) = 7 $ 7sin(31) = S, is equal t 7sin(31). Yu can use yur calculatr t find as a decimal. Runded t the nearest hundredth, sin(31) = 0.52, which yu can use t find. = 7sin(31) = 7(0.52) = 3.64 When a right triangle represents a real-wrld situatin, trignmetric ratis can help yu t answer questins abut missing infrmatin. A flagple has a height f 35 feet. Frm a pint n the grund, the angle f elevatin frm the bttm f the flagple t the tp f the flagple is 52. What is the distance frm the pint t the bttm f the flagple?? Draw and Label a Diagram t Visualize the Prblem Drawing a diagram helps yu t see hw a right triangle mdels the prblem. Use t represent what yu want t find. 1. On a separate sheet f paper, draw and label a right triangle t represent the prblem. 2

3 Slving Prblems with Trignmetry Draw a right triangle s that the bttm leg represents the grund, and the ther leg represents the height f the flagple. Use fr the distance yu want t find c? Use the Diagram t Write a Trignmetric Rati Using a trignmetric rati will help yu t find the missing infrmatin. 2. What trignmetric rati can yu use t help yu find? The sides ppsite and adjacent t the 52 angle are the knwn side and the side yu want t find. The nly trignmetric rati that uses ppsite and adjacent is tangent: tan(52) = 35? Slve the Equatin and Write the Slutin Slve the equatin by getting the variable alne n ne side f the equatin. 3. Find and then write the slutin. Slve the equatin by multiplying bth sides by and then dividing by tan(52) t get alne n ne side f the equatin. tan(52) = 35 $ tan(52) = 35 $ = 35 tan(52) $ = 35 $ =

4 Practice It! Essential Math Skills Answer the fllwing questins. Rund all decimal answers t the nearest hundredth. 1. Yu are standing n the grund 550 yards frm the bttm f a cliff. Frm where yu are standing, the angle f elevatin frm the bttm f the cliff t the tp f the cliff is 8. a. Draw a right triangle t represent this situatin. sin H = cs H = ppsite hyptenuse adjacent hyptenuse ppsite adjacent b. Write a trignmetric rati that can be used t find the height f the cliff. c. Slve yur rati t find the height, in yards, f the cliff. 2. A helicpter is hvering 1,625 feet abve the grund withut mving. The angle f elevatin frm the landing pad t the helicpter is 22. a. Draw a right triangle t represent this situatin. b. Write a trignmetric rati that can be used t find the distance frm the helicpter t the landing pad. c. Slve yur rati t find the distance in feet frm the helicpter t the landing pad. 4

5 Slving Prblems with Trignmetry 3. A 30-meter steel cable is cnnected at the tp f a radi twer t a pint n the grund. Frm this pint, the angle f elevatin frm the bttm f the twer t the tp f the twer is 48. a. Draw a right triangle t represent this situatin. b. Write a trignmetric rati that can be used t find the distance frm the pint t the bttm f the twer. c. Slve yur rati t find the distance in meters frm the pint n the grund t the bttm f the twer. 4. Jnas wants t find the value f this the right triangle. 70c 16 a. Write a trignmetric rati that Jnas can use t find. b. Slve yur rati t find. Build Yur Math Skills Yu can use ther facts abut triangles alng with trignmetric ratis. Remember that the Pythagrean therem can help yu calculate the length f sides. Understanding that the sum f the angles in a trianlge equals 180 can help yu calculate unknwn angles. 5

6 Essential Math Skills 5. A scuba diver is 14.5 meters belw sea level. The angle f decline frm a bat t the diver is 30. a. Draw a right triangle t represent this situatin. b. Write a trignmetric rati that can be used t find the distance frm the bat t the diver. Math Tip An angle f elevatin is an angle frmed frm the hrizntal line up t a pint. An angle f decline is an angle frmed frm the hrizntal line dwn t a pint. c. Slve yur rati t find the distance in meters frm the pint n the grund t the bttm f the twer. 6. In a right triangle, ne f the angles is 70 and the side ppsite the 70 angle is 14 inches. a. Draw a right triangle t represent this situatin. b. Write a trignmetric rati that can be used t find the side that is adjacent t the 70 angle. 14 c. Slve yur rati t shw that the length f the adjacent side is tan(70) inches. 6

Slving Prblems with Trignmetry Check Yur Skills Use yur knwledge f right triangles and trignmetry t answer the fllwing questins. 1. What is the value f cs(62) runded t the nearest hundredth? a. 0.

7 Slving Prblems with Trignmetry Check Yur Skills Use yur knwledge f right triangles and trignmetry t answer the fllwing questins. 1. What is the value f cs(62) runded t the nearest hundredth? a b c d Fr the figure belw, which answer is equal t cs H? 8 H a b c. 8 d A lamp pst casts a shadw 6.8 meters lng. Frm the end f the shadw, the angle f elevatin frm the bttm t the tp f the lamp pst is 51. Hw tall is the lamp pst, runded t the nearest tenth f a meter? a. 8.4 m b. 5.9 m c. 4.3 m d. 6.8 m 7

8 Essential Math Skills 4. Select all equatins that are true fr the figure belw. H 25 7 sin H = 25 cs H = sin H = 7 cs H = A wire is cnnected frm the tp f a twer t a pint n the grund that is 62 feet frm the bttm f the twer. Frm this pint, the angle f elevatin frm the bttm f the twer t the tp f the twer is 75. What is the length f the wire? 62 a. ft tan(75) b. 62 cs(75) ft c. 62 tan(75) ft 62 d. ft cs(75) 6. Fr the figure belw, what is the value f sin H? a. 5 3 b. 5 4 c d. 8 3 H Remember the Cncept Trignmetry can be used t slve prblems invlving right triangles. Yu can find an unknwn side length using trignmetry when yu knw the measure f ne f the acute angles and the length f anther side. 8

9 Answers and Eplanatins Slving Prblems with Trignmetry Using Trignmetry t Find a Distance Practice It! 1a. 550 The distance 550 is the distance alng the grund t the bttm f the cliff, and the smallest angle is 8 (the angle f elevatin t the tp f the cliff). 1b. tan(8) = 550 Use t represent the height f the cliff. Use the tangent rati because the height f the cliff is ppsite the 8 angle, and the distance frm yu t the bttm f the cliff (550 yards) is the adjacent leg. 1c yards 2a. Slve the equatin. tan(8) = 550 Multiply bth sides by tan(8) = Using a calculatr, multiply 550 by tan(8). = c 8c 1,625 The distance 1,625 is the height abve the grund, s this distance and the grund make the legs f the triangle. The elevatin f 22 is the angle frm the grund tward the helicpter. 2b. sin(22) = 1,625 Use t represent the distance frm the landing pad t the helicpter (the hyptenuse). Use the sine rati because the height (1,625 ft) f the helicpter is ppsite the 22 angle, and the distance frm the helicpter t the landing pad is the hyptenuse. 2c feet 3a. Slve the equatin. sin(22) = 1,625 First, multiply by. sin(22) = 1,625 Then, divide by sin(22). = 1,625 sin(22) Using a calculatr, divide 1,625 by sin(22). = c The distance 30 wuld be the hyptenuse f the triangle, because it s the distance frm the pint t the tp f the twer. The angle 48 is the angle frm the pint, lking up t the tp f the twer. 3b. cs(48) = 30 Use t represent the distance frm the pint t the bttm f the twer. Use the csine rati because the distance frm the pint t the bttm f the twer is the adjacent side, and the length f the cable (30 m) is the hyptenuse. 3c meters Slve the equatin. cs(48) = 30 Multiply bth sides by 30: 30 cs(48) = Using a calculatr, multiply 30 by cs(48). = a. sin(70) = 16 Use the sine rati because the side ppsite the 70 angle is, and the hyptenuse is 16. i

10 Essential Math Skills 4b a. Slve the equatin. sin(70) = 16 Multiply bth sides by 16: 16 sin(70) = Using a calculatr, multiply 16 by sin(70). = a. bat 30c 70c diver 14.5 The distance 14.5 is the leg dwn frm the surface f the water t the diver. The angle 30 is the angle frm the bat, lking dwn at the diver. 5b. sin(30) = 14.5 Use t represent the distance frm the bat t the diver. Use the sine rati because the depth f the diver (14.5 m) is ppsite the 30 angle, and the distance frm the bat t the diver is the hyptenuse f the right triangle. 5c meters Slve the equatin. sin(30) = 14.5 Multiply bth sides by. sin(30) = 14.5 Divide bth sides by sin(30). = 14.5 sin(30) Using a calculatr, divide 14.5 by sin(30). = The triangle may be turned a different way, but the leg that is length 14 will be ppsite the angle that is 70. 6b. 14 tan(70) = Use t represent the adjacent side. Use the tangent rati because the ppsite side (14 in.) is given, and is the adjacent side. 6c. Divide by and then divide by tan(70) t shw that is equal t this rati: 14 tan(70) tan(70) = 14 Multiply bth sides by. tan(70) = 14 Divide bth sides by tan(70). = 14 tan(70) Check Yur Skills 1. b Use a calculatr. Press the cs buttn, then 62, and then press enter. Rund the result t the nearest hundredth (2 decimal places). 2. a The csine rati is the side adjacent t H (which is 8) divided by the hyptenuse (which is 17). ii

11 Answers and Eplanatins 3. a. 8.4 m Draw a diagram using fr the height f the lamp pst d. ft cs(75) Draw a diagram and let represent the length f the wire. 51c 6.8 m The sides ppsite and adjacent t the 51 angle are the side yu want t find and the knwn side, which means yu use the tangent rati: tan(51) = 6.8 T find, multiply bth sides by 6.8: 6.8 tan(51) = Using a calculatr, multiply 6.8 by tan(51). = cs H = 25 7 The hyptenuse is 25, the ppsite side is 7, and the adjacent side is. This triangle has the fllwing trigmetric ratis: sin H = 25 7 cs H = c 62 ft Use the csine rati because 62 is the adjacent side and is the hyptenuse: cs(75) = 62 Multiply bth sides by : cs(75) = 62 Sve fr by dividing bth sides by cs(75): = 62 cs(75) 6. b. 5 4 T find sin H, yu need t find the ppsite side, which is unknwn. Use the Pythagrean therem using a fr the length f the unknwn side: a = 5 2 Calculate the epnents: a = 25 Subtract 9 frm each side: a 2 = 16 Take the square rt t find the value f a. The value must be psitive, since it s the measurement f a length. a = 16 = 4 The ppsite side is 4, and the hyptenuse is 5. Therefre, sin H = 5 4. iii

UNIT 7 RIGHT ANGLE TRIANGLES

UNIT 7 RIGHT ANGLE TRIANGLES Assignment Title Wrk t cmplete Cmplete Cmplete the vcabulary wrds n Vcabulary the attached handut with infrmatin frm the bklet r text. 1 Triangles Labelling Triangles 2 Pythagrean