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Name Date Class Practice B LESSON 8-2 Trigonometric Ratios Use the figure for Exercises 1–6. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth. 1. sin A 2. cos B ! # 3. tan B 7 ; 0.28 ___ 7 ; 0.28 ___ 25 4. sin B " 24 ; 3.43 ___ 7 25 5. cos A 6. tan A 24 ; 0.96 ___ 24 ; 0.96 ___ 25 7 ; 0.29 ___ 25 24 Use special right triangles to write each trigonometric ratio as a simplified fraction. 1 __ 7. sin 30 8. cos 30 2 3 ___ 3 10. tan 30 11. cos 45 3 ___ 2 2 ___ 2 9. tan 45 1 12. tan 60 3 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 0.90 13. sin 64° 14. cos 58° 0.53 15. tan 15° 0.27 Find each length. Round to the nearest hundredth. 9 16. ( 17. 18. ' + 0.8 mi 12.2 in. 8 55° 41° XZ 14.03 in. 19. 36° 3 100 cm ) : 4 HI 57.36 cm 72° 8.68 km Copyright © by Holt, Rinehart and Winston. All rights reserved. # % 2 ft & 51° $ 2 ST 31 yd 21. , 0.36 mi KM $ 20. % 5.1 km 27° - EF 95.41 yd 12 DE 3.18 ft Holt Geometry Name Date Class Name Practice A LESSON 8-2 8-2 In Exercises 1–3, fill in the blanks to complete each definition. Then use side lengths from the figure to complete the indicated trigonometric ratios. � � � � Trigonometric Ratios Use the figure for Exercises 1–6. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth. � � opposite 1. The sine (sin) of an angle is the ratio of the length of the leg hypotenuse . the angle to the length of the 1. sin A c b b a 5. cos L � _3_ ; 0.60 10 � 6. tan M _4_ ; 0.80 8. cos 30� 2 � �3 ___ 3 10. tan 30� 6 11. cos 45� �3 ___ 2 � �2 ___ 2 9. tan 45� 1 12. tan 60� �3 � � 8 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. _4_ ; 1.33 5 5 24 � _1_ 7. sin 30� Use the figure for Exercises 4–6. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth. 7 ; 0.29 ___ 25 Use special right triangles to write each trigonometric ratio as a simplified fraction. b tan B � ___ a tan A � ___ 24 ; 0.96 ___ 25 opposite 3. The tangent (tan) of an angle is the ratio of the length of the leg adjacent the angle to the length of the leg to the angle. 7 6. tan A 24 ; 0.96 ___ c 24 ; 3.43 ___ 5. cos A a cos B � ___ cos A � ___ c 3. tan B 25 4. sin B adjacent � � �� 7 ; 0.28 ___ 25 2. The cosine (cos) of an angle is the ratio of the length of the leg hypotenuse to the angle to the length of the . � �� � 2. cos B 7 ; 0.28 ___ b sin B � ___ a sin A � ___ c Class Practice B LESSON Trigonometric Ratios 4. sin L Date 3 0.90 13. sin 64° 14. cos 58° 0.53 0.27 15. tan 15° Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 8. cos 47� 7. sin 33� 9. tan 81� Find each length. Round to the nearest hundredth. 6.31 0.68 0.54 16. � 62° � � � 12.46 m 12. � 5 ft � � 27° 13. The glide slope is the path a plane uses while it is landing on a runway. The glide slope usually makes a 3� angle with the ground. A plane is on the glide slope and is 1 mile (5280 feet) from touchdown. Use the tangent ratio and a calculator to find EF, the plane’s altitude, to the nearest foot. 11 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name 8-2 � � ���� Date Class Holt Geometry 4. 9 ft 78° � � � x � ___ 1 � ____ � � x� 2 �2 � 2 � ___ � 2 13 Copyright © by Holt, Rinehart and Winston. All rights reserved. � � � � 60° 45° � �� 2 2� � 45° � � � 30° � � �� 3 � � �2 . So sin 45� � ___ 2 Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 1. sin K � 2. cos H 15 � 0.88 ___ � 17 8 � 15 � 15 � 0.88 ___ 17 17 3. cos K 4. tan H 8 � 0.47 ___ 8 � 0.53 ___ 17 15 Use a special right triangle to write each trigonometric ratio as a fraction. 6. tan 45� 5. cos 45� � �2 ___ _1_ � 1 1 2 7. sin 60� 44.1 ft 8. tan 30� � � �3 ___ 15 km 15.18 km � leg adjacent to �A leg opposite �Q sin 45� � sin Q � ______________ hypotenuse 75° �3 ___ 2 6.9 cm 6.78 cm � � You can use special right triangles to write trigonometric ratios as fractions. Use the formula you developed in Exercise 5 to find the missing side length in each triangle. Round to the nearest hundredth. Copyright © by Holt, Rinehart and Winston. All rights reserved. leg opposite �A � leg opposite �A tan A � ________________ � _4_ � 1.33 3 leg adjacent to �A 7 in. 44.1 ft � hypotenuse leg adjacent to �A cos A � ________________ � _3_ � 0.6 5 hypotenuse 2 2 2 2 2 2 2 gives b � 2bx � x � y � a . Substituting, b � 2bx � c � a . x _ _ But cos A � c, so x � c cos A. Another substitution gives 2 2 2 a � b � c � 2bc cos A. 4 km 85° Holt Geometry Trigonometric Ratios 2 2 2 It also shows that (b � x) � y � a . Expanding the latter equation 55° Class Trigonometric Ratios 12.05 in 8. Date Reteach leg opposite �A sin A � ______________ � _4_ � 0.8 5 hypotenuse Possible answer: The Pythagorean Theorem shows that x 2 � y 2 � c 2. 7. 3.18 ft DE � � 2 28.39 ft 7.7 cm � 51° 95.41 yd 12 � � 5. The law of cosines is a formula to find the length of the third side of any triangle given the lengths of two sides and the measure of the � � � included angle. (Note: The law of cosines applies to any triangle, but deriving it for an obtuse triangle requires more knowledge than � ����� � you have learned so far.) In the figure, suppose the lengths b and c � and the measure of �A are known. Develop the law of cosines to find a. Explain your answer. (Hint: Use the cosine function and the Pythagorean Theorem.) 6. EF Name 7.1 in. 2 18.66 m � 2 ft � 8.68 km 73° 7.6 m 2 31 yd � Copyright © by Holt, Rinehart and Winston. All rights reserved. Use the formula you developed in Exercise 1 to find the area of each triangle. Round to the nearest hundredth. 43° 21. � 72° � 0.36 mi KM � ST 8-2 11 ft 20. � 27° 277 feet Possible answer: Draw an altitude from �B and call its length h. Then _ sin A � _h c , so h � c sin A. The formula for the area of a triangle is Area � _1_ base � height. Substitution gives Area � _1_ bc sin A. 2 2 35° 57.36 cm � LESSON 3. HI 5.1 km Trigonometric Ratios 7.2 m 36° � 1. Given the lengths of two sides of a triangle and the measure of the included angle, the area of the triangle can be found. In the figure, suppose the lengths b and c and the measure of �A are known. Develop a formula for finding the area. Explain your answer. (Hint: Draw an altitude.) 2. 0.8 mi � �� Practice C LESSON 14.03 in. � 2.55 feet RS � 100 cm � � 19. � � 19.70 mm QP � 55° 41° XZ 38° 25 mm 11 m BD � 11. 18. � 12.2 in. Use a calculator and trigonometric ratios to find each length. Round to the nearest hundredth. 10. 17. � 3 22.83 ft Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 53 14 Holt Geometry Holt Geometry