Ready To Go On? Skills Intervention 13-1 Right-Angle Trigonometry

Transcription

1 Find these vocabulary words in Lesson 13-1 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 13-1 Right-Angle Trigonometry trigonometric function sine cosine tangent cosecant secant cotangent Finding Values of Trigonometric Functions In a right triangle, sin opp hyp, cos adj hyp, tan opp adj A. Find the value of the sine function for the triangle shown. sin opp hyp hyp sin opp hyp 8 sin opp hyp The side opposite is the side that does not touch the angle. The hypotenuse is the side opposite from the right angle. Solve. B. Find the value of the cosine function for the triangle shown. cos adj hyp hyp cos adj The hyp cos adj hyp The side adjacent is the side that touches the angle. hypotenuse is the side opposite from the right angle. Solve. C. Find the value of the tangent function for the triangle shown. tan opp adj tan opp adj The opposite and the adjacent are the two shortest sides. D. Find the value of the cosecant function for the triangle shown. 1 csc sin 1 csc csc.13 8 cm 17 cm 0 15 cm 07 Holt Algebra

2 Ready To Go On? Problem Solving Intervention 13-1 Right-Angle Trigonometry A trigonometric ratio compares the length of two sides of right triangles. Foresters use a simple device called a clinometer to determine the height of trees by standing a certain distance from the base and measuring the angle to the top. A forester obtains the following measurements a. Angle to top of tree: 30 b. Distance from base of tree: 50 m c. Height of the forester s eyes above ground: 1.68 m What is the height of the tree? 30 Understand the Problem Eamine the figure. 1. Is there a right triangle?. What are the sides of the triangle? 1.68 m 50 m 3. Which side do you need to find? 4. What should you do with the height of the forester s eyes above ground? Make a Plan 5. Is the unknown the opposite side, adjacent side, or hypotenuse? 6. Is the known side the opposite side, adjacent side, or hypotenuse? 7. Which trig function includes the knowns and the unknowns from Eercise 5 Solve and 6? 8. Substitute known values into the tangent ratio and solve: 9. Add the height of eye level to find the height of the tree. Look Back 10. You can check your solution by using the formula for a right triangle. The sides are in proportion as shown. Complete: tan 30 opp opp 1. Solve for. 08 Holt Algebra

3 Ready To Go On? Skills Intervention 13- Angles of Rotation Find these vocabulary words in Lesson 13- and the Multilingual Glossary. Vocabulary standard position initial side terminal side angle of rotation coterminal angles reference angle Drawing Angles in Standard Position Draw an angle with the given measure in standard position. A. 67 B. 15 The angle is positive so The angle is positive so rotate the terminal side rotate the terminal side counterclockwise.. y y C. 50 The angle is negative so rotate the side clockwise. y Finding Coterminal Angles Find a positive angle coterminal with each given angle. A. 35 To find a positive coterminal angle, B. 135 To find a positive coterminal angle, add 360. add Finding Reference Angles Find the measure of the reference angle for each given angle. C. 35 To find a positive coterminal angle, 35 The reference angle is the positive acute angle formed by the terminal side of and the -ais. A. 135 B. 15 The reference angle The reference angle. y. y 360. C. 50 The reference angle. y 09 Holt Algebra

4 Ready To Go On? Problem Solving Intervention 13- Angles of Rotation In ocean navigation, the course of a vessel is measured as a clockwise angle from true north. This angle is also called the bearing. Suppose a vessel leaves an oil terminal and sails for three hours with a heading of 35. The captain then orders the vessel to turn to a new bearing, 158. Through what angle did the ship turn? Understand the Problem 1. What is the initial bearing?. What is the new bearing? N N Make a Plan 3. Draw a ray from home port due north or Rotate this ray 35 clockwise. This is the ship s bearing. 5. Mark a point along the new ray representing the position of the vessel where it makes its turn. 6. Draw a ray due north from this point. 7. Rotate this ray 158 clockwise and draw a new ray representing the vessel s new heading. 8. Identify the angle through which the vessel turned. Home port 35 Path of vessel 158 Solve 9. What is the angle through which the vessel turned? Look Back 10. You can check your solution by using a protractor to measure the angle between the original bearing and the new bearing. N What angle do you measure? How does this compare to your answer in Eercise 7? Home port 35 Path of vessel 10 Holt Algebra

5 Ready To Go On? Skills Intervention 13-3 The Unit Circle Find these vocabulary words in Lesson 13-3 and the Multilingual Glossary. Vocabulary radian unit circle Converting Between Degrees and Radians Find the equivalent radian measure for each angle. To convert from radians to degrees, multiply by 180. A B C. 138 D. 0 0 E. 70 F Find the equivalent degree measure for each angle. To convert from degrees to radians, multiply by A. B C D E. 3 5 F Using the Unit Circle to Evaluate Trigonometric Functions The point r, on the unit circle, has coordinates (, y); cos sin y. Use the unit circle to find the eact value for each angle. A. sin 60 C. cos 5 6 B. sin The Unit Circle II III I IV Holt Algebra

6 Ready To Go On? Problem Solving Intervention 13-3 The Unit Circle A radian is a unit of angle measure based on arc length. A well known bicycle racer finished a stage of the Tour de France with an average speed of 49 km/h, or 13.3 m/s. If the outside diameter of the bicycle wheel is 0.70 m, what is the angular speed of the wheel in radians per second and degrees per second? Understand the Problem 1. If the linear speed of the bicycle is 13.3 m/s, what is the linear speed of a point on the circumference of the tire?. How far does the bicycle travel as the wheel makes one rotation? 3. How far does the bicycle travel in one second? Make a Plan 4. What is the radius of the wheel? 5. What is the circumference of the wheel? 6. How many times does the wheel revolve as the bicycle moves 13.3 m? Solve 7. Convert revolutions to radians. 8. Convert radians to degrees. 9. What is the angular speed of the wheel in radians per second and degrees per second? Look Back 10. You can check your solution by working backwards revolution revolutions 6 revolutions. m revolution m 11. How do your answers to Eercise 9 and Eercise 10 compare? 1 Holt Algebra

7 Ready To Go On? Skills Intervention 13-4 Inverses of Trigonometric Functions Find these vocabulary words in Lesson 13-4 and the Multilingual Glossary. Vocabulary inverse sine function inverse cosine function inverse tangent function Writing Inverse Trigonometric Functions Rewrite each of the following equations in a form that uses the inverse trigonometric functions. A. sin Write as an inverse sine function: si n 1 B. cos 3 Write as an inverse cosine function: co s 1 C. tan y 3 Write as an inverse tangent function: y ta n 1 Solving Trigonometric Equations Find the values of in the interval that satisfies each equation. A. si n 1 0 Use the inverse sine function on your calculator. 0, B. ta n 1 1 Use the inverse tangent function on your calculator. 45, Find all possible values of that satisfy each equation. D. ta n 1 0 Use the unit circle to find values for 0. 0, Net, find angles that are coterminal with n, where n 0, 1,, 3, 4,... C. si n 1 3 Use the inverse sine function on your calculator., 300 E. ta n 1 (1) Use the unit circle to find values for , Net, find angles that are coterminal with 4 Evaluating Inverse Trigonometric Ratios State the domain and range of each inverse trigonometric functions. A. Si n 1 is restricted to quadrants I and IV. Domain: 1 Range: n, 7 n, n 1,, 3, 4,... B. Co s 1 is restricted to quadrants I and II. Domain: 1 Range: 13 Holt Algebra

8 Ready To Go On? Problem Solving Intervention 13-4 Inverses of Trigonometric Functions You can use inverse trigonometric functions to find unknown measures. During a training eercise, a fire truck is attempting to rescue another firefighter from the roof of a seven story building. The roof is 84 feet from the ground. The truck is parked so that the base of the ladder is 45 feet from the building and the ladder is etended to the roof. The base of the ladder is 8 feet from the ground. The optimum angle for the ladder is 60. What is the angle of this ladder? Understand the Problem 1. Draw the appropriate right triangle and add dimensions to the sides. Be sure to allow for the height of the fire truck.. What is known? feet 3. What is unknown? Make a Plan 0 4. Which trigonometric function is related to the two known sides? 45 feet 8 feet 5. Since the angle is unknown but the two sides are known, do you need to use the tangent function or its inverse to solve the problem? 6. Write the appropriate trigonometric function in proper form: ta n 1 Solve 7. Substitute the appropriate values into the function in Eercise 6. opp ta n 1 ta n 1 76 ta n 1 adj 8. Use a calculator to find the corresponding angle. 9. The angle the ladder makes is. Look Back 10. Use a calculator to find the tangent of 60 and work backwards to check your answer. tan 60 opp adj adj opp feet How does your answer from Eercise 10 compare to 45 feet? 14 Holt Algebra

9 13-1 Right-Angle Trigonometry Find the values of the three trigonometric functions for Ready To Go On? Quiz Use a trigonometric function to find the value of Angles of Rotation Draw an angle with the given measure in standard position y y Point P is a point on the terminal side of in standard position. Find the eact value of the si trigonometric functions for. 11. P(3, 4) 1. P(8, 15) 15 Holt Algebra

10 Ready To Go On? Quiz continued 13. The angle of elevation to the top of a building whose base is 00 feet away is 30. The angle of elevation to the top of a radio tower on top of the building is 45. What is the height of the tower above the roof? 13-3 The Unit Circle Convert each measure from degrees to radians or from radians to degrees Use the unit circle to find the eact value of each trigonometric function. 16. cos sec Inverses of Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. 18. co s ta n When the sun is at an angle of 6 above the horizon, a flagpole casts a shadow directly down a 10 slope. The length of the shadow is 4.4 m. What is the height of the flagpole? sunlight m shadow of pole Holt Algebra

11 Ready To Go On? Enrichment Right Triangles with Sides in Integral Ratios There is a legend that the ancient Egyptians knew how to construct a right angle by using a right triangle. Whether they did or not, many craft workers have known how to mark off three units along one board and four along another to check whether the boards are at a right angle. Answer each question. 1. Show that a triangle whose sides are in the proportions 3:4:5 is a right triangle.. Are there any other right triangles with sides in integer proportions? Test several combinations to find out? 3. Integers with the property a b c are known as Pythagorean triples. Test the following combinations to see if they are Pythagorean triples: a b c a b c Yes/No How many Pythagorean triples do you think eist? 5. Pythagorean triples can be generated by the following procedure. Let m and n be any positive integers, with n m. Then a n m, b nm, c n m. The proof is simple. Substitute the epressions into a b and simplify. a b n m 17 Holt Algebra

12 13B Ready To Go On? Skills Intervention 13-5 The Law of Sines For computation, the angles of a triangle are assigned upper case letters and the opposite sides are assigned lower case letters. Using this system, the area of any triangle can be calculated. A Area 1 bc sin A, where A can represent any angle and b and c the adjacent sides c b Determining the Area of a Triangle Find the area of each triangle. Round to the nearest tenth. B a C A. B B. B c = 9.5 A 80 b = a 100 C 9.3 A C Area 1 bc sin A Area 1 bc sin A 1 sin (9.3) sin Using the Law of Sines Solve the triangle for the unknown sides and angles. A Find the angle whose opposite side is known: sin B sin C b c sin B sin 10 6 C a B sin 10 sin B 16 B si n 1 (0.35) 19.0 Find the remaining angle: ma mb mc 180 ma 180 mb mc 180 Find the remaining side: sin A sin C a c a c sin A 16 sin sin 18 Holt Algebra

13 13B Ready To Go On? Problem Solving Intervention 13-5 The Law of Sines You can use the Law of Sines to solve for unknown measures in a triangle. Two forest rangers in two different towers spot a small fire. They must determine the position of the fire. The two towers are 10 miles apart and are due east and west of each other. The closest access point is a river that flows between them. One ranger determines the location of the fire to be 45 east of north. The other sees it 34 west of north. Use appropriate trigonometric functions to solve the triangle and locate the fire. Understand the Problem 1. What is known? N C N. What do you need to determine? b a Can you use the Law of Sines to find the missing information? 4. What is the first step? A D c 10 miles B Make a Plan 5. Find the missing angle. mc Write the Law of Sines to solve for a or b 7. Rearrange the epression and solve for the missing sides. a c sin A sin C sin sin miles b c sin B sin C sin sin miles Solve 8. Use the cosine function and either ACD or BCD to find h, the height of the triangle. h b sin A miles sin miles Look Back 9. Compare the value of h to the values of a and b. Is your answer to Eercise 8 less than that of a or b? Is your answer reasonable? 19 Holt Algebra

14 13B For computation, the angles of a triangle are assigned upper case letters and the opposite sides are assigned lower case letters. The Law of Cosines can be written as c a b ab cos C. Any angle can be assigned the letter C, so the law can be written using any angle, as long as the same relationship between sides and angles is preserved. Determining Which Law to Use The measures of three parts of a triangle are given below. Determine whether the Law of Sines or the Law of Cosines should be used to solve the triangle. A. a 3, b 4, c 5 Since no given, the Law of are should be used to find one angle. B. a 11, b 1, C 4 Two and the included are known, so Law of used. should be Using the Law of Cosines Solve each triangle. Round values to the nearest tenth. A. a, b 3, c 4 Step 1 First use the Law of Cosines. cos C a b c ab cos C Ready To Go On? Skills Intervention 13-6 The Law of Cosines C co s 1 Step Now use the Law of Sines. sin A sin C a c sin A a sin C sin A si n 1 Step 3 Two angles are known. Find the third angle by subtraction. B 180 A C 180 The triangle is completely solved. B B. A 55, b 1, c 7 c A C. a 8.5, A 41, C 10 a Since an angle and the side are given, Law of should be used. Step 1 First use the Law of Cosines. a b c bc cos A a c a cos Step Now use the Law of Sines. sin C sin A c a sin C c sin a A sin (55) C si n 1 Step 3 Two angles are known. Find the third angle by subtraction B 180 A C 180 The triangle is completely solved. b C 0 Holt Algebra

15 13B Ready To Go On? Problem Solving Intervention 13-6 The Law of Cosines You can solve a triangle for which SAS or SSS information is given using the Law of Cosines. In college fast pitch softball, the diamond is a square 60 feet on a side. The front edge of the pitcher s plate is 43 feet from the back point of home plate. How far is it from the front edge of the pitcher s plate to the outside of first base? Understand the Problem 1. Do home plate, first base, and the pitcher s HOME RUN FENCE plate form a right triangle?. Is this triangle SSA, ASA, SAS, or AAS? FOUL POLE 10' W FOUL LINE WA RNING TRACK 60' GRASS LINE 90 60' 190' 5' FOUL LINE FOUL POLE 3. Redraw the triangle and label the sides and angles. 4. Will the Law of Cosines provide the information needed? Measured Outside Edge to Ape 60' 43' 45 60' Make a Plan 5. Using the diagram in Eercise 3, write the Law of Cosines in appropriate form. 6. List each of the known values: a, b, C Solve 7. Substitute the known values into the equation. c cos c 8. How far is it from the front edge of the pitcher s plate to the outside of first base? c ft Look Back 9. Since the front edge of the pitcher s plate is 43 feet from the back point of home plate, does your answer in Eercise 8 seem reasonable? Eplain. 1 Holt Algebra

16 13B Ready To Go On? Quiz 13-5 The Law of Sines Find the area of each triangle. Round to the nearest tenth Solve each triangle. Round to the nearest tenth. 3. C 4. b A 9 6 a 11.5 B C b B 37 c A 5. Two observers spot a hot air balloon over Albuquerque, New Meico. The balloon is on a direct line between the observers, who are.8 km apart. One observer measures the angle of elevation of the balloon to be 33 and the other measures the angle of elevation to be 37. What is the altitude above ground of the balloon? 6. Walden Street and Waverly Avenue intersect at an angle of 85. In the acute angle between the two streets is a small triangular park. The length of the park boundary along Waverly Avenue is 5 m. The length along Walden Street is 35 m. A landscape designer wants to seed the park with Prairie Buffalo grass. How many square meters of park must be seeded? Walden Street 35 m 85 5 m Waverley Avenue Holt Algebra

17 13B Ready To Go On? Quiz continued 7. Two houses sit along the southern shore of a lake. The distance between the houses is 150 m. From one house, a green light can be seen at the end of a dock across the lake. The direction to the light from this house is 8 east of north. From the other house the light appears to be 48 west of north. What is the distance from each house to the light? N 8 green light 48 N 150 m 13-6 The Law of Cosines Use the given measurements to solve each triangle. Round to the nearest tenth 8. a 5, b 6, c 7 9. b 7, c 10, A A 53, b 191, c a 9, c 5, B The landscape designer in Question 6 also wants to plant a border along the perimeter of the park on all three sides. What is the perimeter? Walden Street 35 m 85 5 m Waverley Avenue 3 Holt Algebra

18 13B Ready To Go On? Enrichment Law of Sines The Laws of Sines, the Law of Cosines, and the area of a triangle formula can be used to find the area of any irregular polygon. Answer each question. C Pick a verte and draw line segments from that verte to each of the others to separate the polygon into triangles.. Use the Law of Cosines to find the side opposite from the 37 angle. c c cos A 3.5 D 6.5 B 3. Use the Law of Cosines to find the other opposite angle. cos A 6. 5 (3.5)(6.5) A co s 1 4. Now find the area of each triangle. Area 1 bc sin A 1 (6.8) sin Area 1 bc sin A 1 (6.5) sin Use the Law of Sines to find CBD. sincbd 4. 1 sin 37 CBD si n Use the Law of Sines again to find DBA. sindba sin 35.9 DBA si n sin Use addition and subtraction to find the remaining angles. CBA 9.3, CDA Holt Algebra