13.4 Problem Solving with Trigonometry

Transcription

1 Name lass ate 13.4 Problem Solving with Trigonometr Essential Question: How can ou solve a right triangle? Resource Locker Eplore eriving an rea Formula You can use trigonometr to find the area of a triangle without knowing its height. Suppose ou draw an altitude _ to side _ of. Then write an equation using a trigonometric ratio in terms of, the height h of, and the length of one of its sides. b c b h c a a Solve our equation from Step for h. Houghton Mifflin Harcourt Publishing ompan omplete this formula for the area of in terms of h and another of its side lengths: rea = 1_ Substitute our epression for h from Step into our formula from Step. Reflect 1. oes the area formula ou found work if is a right angle? Eplain. Module Lesson 4

2 . Suppose ou used a trigonometric ratio in terms of, h, and a different side length. How would this change our findings? What does this tell ou about the choice of sides and included angle? Eplain 1 Using the rea Formula rea Formula for a Triangle in Terms of its Side Lengths The area of with sides a, b, and c can be found using the lengths of two of its sides and the sine of the included angle: rea = 1 bc sin, rea = 1 ac sin, or rea = 1 ab sin. a c b You can use an form of the area formula to find the area of a triangle, given two side lengths and the measure of the included angle. Eample 1 Find the area of each triangle to the nearest tenth. 3. m m Let the known side lengths be a and b. a = 3. m and b = 4.7 m Let the known angle be. m = 14 Substitute in the formula rea = 1_ ab sin. rea = 1_ (3.)(4.7)sin 14 Evaluate, rounding to the nearest tenth. rea 4.6 m Houghton Mifflin Harcourt Publishing ompan Module Lesson 4

3 In EF, E = 9 in., F = 13 in., and m = 7. Sketch EF and check that is the included angle. Write the area formula in terms of EF. rea = 1_ (E) ( ) sin Substitute in the area formula. rea = 1_ ( ) ( ) sin Evaluate, rounding to the nearest tenth. rea i n. Your Turn Find the area of each triangle to the nearest tenth mm 34 1 mm Houghton Mifflin Harcourt Publishing ompan 4. In PQR, PQ = 3 cm, QR = 6 cm, and m Q = P Q 3 cm R 6 cm Module 13 7 Lesson 4

4 Eplain Solving a Right Triangle Solving a right triangle means finding the lengths of all its sides and the measures of all its angles. To solve a right triangle ou need to know two side lengths or one side length and an acute angle measure. ased on the given information, choose among trigonometric ratios, inverse trigonometric ratios, and the Pthagorean Theorem to help ou solve the right triangle. shelf etends perpendicularl 7 in. from a wall. You want to place a 9-in. brace under the shelf, as shown. To the nearest tenth of an inch, how far below the shelf will the brace be attached to the wall? To the nearest degree, what angle will the brace make with the shelf and with the wall? 7 in. Shelf Find. Wall race 9 in. Use the Pthagorean Theorem to find the length + = of the third side. Substitute 7 for and 9 for. 7 + = 9 Find the squares = 81 Subtract 49 from both sides. = 3 Find the square root and root..7 Find m and m. Use an inverse trigonometric ratio to find m. You know the lengths of the adjacent side and the hpotenuse, so use the cosine ratio. Write a cosine ratio for. cos = Write an inverse cosine ratio. m = c os -1 ( ) Evaluate the inverse cosine ratio and round. m and are complementar. m + m = 9 Substitute for m. + m 9 Subtract from both sides. m Houghton Mifflin Harcourt Publishing ompan Reflect. Is it possible to find m before ou find m? Eplain. Module Lesson 4

5 Your Turn building casts a 33-m shadow when the Sun is at an angle of 7 to the vertical. How tall is the building, to the nearest meter? How far is it from the top of the building to the tip of the shadow? What angle does a ra from the Sun along the edge of the shadow make with the ground? 7 F uilding 6. Use a trigonometric ratio to find the distance EF. 33 m E Shadow 7. Use another trigonometric ratio to find the distance F. 8. Use the fact that acute angles of a right triangle are complementar to find m. Houghton Mifflin Harcourt Publishing ompan Eplain 3 Solving a Right Triangle in the oordinate Plane You can use the distance formula as well as trigonometric tools to solve right triangles in the coordinate plane. Eample 3 Solve each triangle. Triangle has vertices (-3, 3), (-3, -1), and (4, -1). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Plot points,, and, and draw. Find the side lengths: = 4, = 7 Use the distance formula to find the length of _. = (4 - (-3) ) + (-1-3) = Find the angle measures: _ _, so m = 9. Use an inverse tangent ratio to find -1 m = tan ( _ ) = tan -1 ( 4_ 7) 3. and are complementar, so m 9-3 = 6. Module Lesson 4

6 Triangle EF has vertices (-4, 3), E (3, 4), and F (, ). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Plot points, E, and F, and draw EF. F appears to be a right angle. To check, find the slope _ - 3 of F : _ = _ = ; - slope of _ - : = _ = _ ; - 3 so m F =. Find the side lengths using the distance formula: E = - (3 ) ( ) = = F = - ( ) ( + - 3) = = - ( ) ( + - 4) Use an inverse sine ratio to find m. =, =, = ( m = si n -1 EF _ ) = si n -1 ( _ ) = and are complementar, so m = 9 - =. Reflect 9. How does the given information determine which inverse trigonometric ratio ou should use to determine an acute angle measure? Houghton Mifflin Harcourt Publishing ompan Module Lesson 4

7 Your Turn 1. Triangle JKL has vertices J (3, ), K (-3, ), and L (, 1). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Elaborate 11. Would ou use the area formula ou determined in this lesson for a right triangle? Eplain. 1. iscussion How does the process of solving a right triangle change when its vertices are located in the coordinate plane? Houghton Mifflin Harcourt Publishing ompan 13. Essential Question heck-in How do ou find the unknown angle measures in a right triangle? Module Lesson 4

8 Evaluate: Homework and Practice Find the area of each triangle to the nearest tenth cm 3. cm Online Homework Hints and Help Etra Practice 1.. In PQR, PR = 3 mm, QR = 39 mm, and m R = 163. Solve each right triangle. Round lengths to the nearest tenth and angles to the nearest degree cm.7 cm 4. F 6 6 m. Right PQR with _ PQ E _ PR, QR = 47 mm, and m Q = Houghton Mifflin Harcourt Publishing ompan Module Lesson 4

9 Solve each triangle. Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 6. Triangle with vertices (-4, 4), (3, 4), and (3, -) 7. Triangle JKL with vertices J (-3, 1), K (-1, 4), and L (6, ) Houghton Mifflin Harcourt Publishing ompan 8. Triangle PQR with vertices P (, ), Q (, 3), and R (-4, -) Module Lesson 4

10 9. Surveing plot of land is in the shape of a triangle, as shown. Find the area of the plot, to the nearest hundred square ards. 14 d 18 7 d 1. Histor drawbridge at the entrance to an ancient castle is raised and lowered b a pair of chains. The figure represents the drawbridge when flat. Find the height of the suspension point of the chain, to the nearest tenth of a meter, and the measures of the acute angles the chain makes with the wall and the drawbridge, to the nearest degree.. m hain Wall 3. m drawbridge 11. uilding For safet, the angle a wheelchair ramp makes with the horizontal should be no more than 3.. What is the maimum height of a ramp of length 3 ft? What distance along the ground would this ramp cover? Round to the nearest tenth of a foot. 3 ft 3. Houghton Mifflin Harcourt Publishing ompan Module Lesson 4

11 1. Multi-Step The figure shows an origami crane as well as a stage of its construction. The area of each wing is shown b the shaded part of the figure, which is smmetric about its vertical center line. Use the information in the figure to find the total wing area of the crane, to the nearest tenth of a square inch.. in. 4 F 4. in. E 3.7 in.. Houghton Mifflin Harcourt Publishing ompan Image redits: Jules Kitano/iStockPhoto.com 13. Right triangle XYZ has vertices X (1, 4) and Y (, -3). The verte Z has positive integer coordinates, and XZ =. Find the coordinates of Z and solve XYZ; give eact answers. Module Lesson 4

12 14. ritique Reasoning Shania and Pedro are discussing whether it is alwas possible to solve a right triangle, given enough information, without using the Pthagorean Theorem. Pedro sas that it is alwas possible, but Shania thinks that when two side lengths and no angle measures are given, the Pthagorean Theorem is needed. Who is correct, and wh? 1. esign The logo shown is smmetrical about one of its diagonals. Find the angle measures in E, to the nearest degree. (Hint: First find an angle in, E or EF) Then, find the area of E, without first finding the areas of the other triangles. 6 mm F E 3 mm 3 mm 16. Use the area formula for obtuse in the diagram to show that if an acute angle and an obtuse angle are supplementar, then their sines are equal. a b c h Houghton Mifflin Harcourt Publishing ompan Module Lesson 4

13 H.O.T. Focus on Higher Order Thinking 17. ommunicate Mathematical Ideas The HL ongruence Theorem states that for right triangles and EF such that and are right angles, _ EF, and E, EF. Eplain, without formal proof, how solving a right triangle with given leg lengths, or with a given side length and acute angle measure, shows that right triangles with both legs congruent, or with corresponding sides and angles congruent, must be congruent. F E 18. Persevere in Problem Solving Find the perimeter and area of, as eact numbers. Then, find the measures of all the angles to the nearest degree. Houghton Mifflin Harcourt Publishing ompan 19. nalze Relationships Find the area of the triangle using two different formulas, and deduce an epression for sinθ. θ θ Module Lesson 4

14 Lesson Performance Task Ever molecule of water contains two atoms of hdrogen and one atom of ogen. The drawing shows how the atoms are arranged in a molecule of water, along with the incredibl precise dimensions of the molecule that phsicists have been able to determine. (1 pm = 1 picometer = 1-1 m) 1. raw and label a triangle with the dimensions shown pm O pm. Find the area of the triangle in square centimeters. Show our work. H H 3. Find the distance between the hdrogen atoms in centimeters. Eplain our method. Houghton Mifflin Harcourt Publishing ompan Module Lesson 4