sin A cos A Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 73 of 182

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1 UNIT 3: RIGHT TRIANGLE TRIGONOMETRY This unit investigates the properties of right triangles. The trigonometric ratios sine, cosine, an tangent along with the Pythagorean theorem are use to solve right triangles in applie problems. The relationship between the sine an cosine of complementary angles is ientifie. Right Triangle Relationships MGSE9-12.G.SRT.6 Unerstan that by similarity, sie ratios in right triangles are properties of the angles in the triangle, leaing to efinitions of trigonometric ratios for acute angles. MGSE9-12.G.SRT.7 Eplain an use the relationship between the sine an cosine of complementary angles. MGSE9-12.G.SRT.8 Use trigonometric ratios an the Pythagorean theorem to solve right triangles in applie problems. KEY IDEAS 1. The trigonometric ratios sine, cosine, an tangent are efine as ratios of the lengths of the sies in a right triangle with a given acute angle measure. These terms are usually seen abbreviate as sin, cos, an tan. sin θ = length of ajacent sie cos θ = tan θ = length of ajacent sie sin A = length of ajacent sie cos A = tan A = length of ajacent sie 2. The two acute angles of any right triangle are complementary. As a result, if angles P an Q are complementary, sin P = cos Q an sin Q = cos P. 3. When solving problems with right triangles, you can use both trigonometric ratios an the Pythagorean theorem (a 2 + b 2 = c 2 ). There may be more than one way to solve the problem, so analyze the given information to help ecie which metho is the most efficient. Important Tip sin A The tangent of angle A is also equivalent to cos A. Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents Page 73 of 182

2 REVIEW EXAMPLES 1. Triangles ABC an DEF are similar. a. Fin the ratio of the sie opposite angle B to the hypotenuse in ABC. b. What angle in DEF correspons to angle B? c. Fin the ratio of the sie opposite angle E to the hypotenuse in DEF.. How oes the ratio in part (a) compare to the ratio in part (c)? e. Which trigonometric ratio oes this represent? Solution: a. AC is opposite angle B. BC is the hypotenuse. The ratio of the sie opposite angle B to the hypotenuse in ABC is 8 10 = 4 5. b. Angle E in DEF correspons to angle B in ABC. c. DF is opposite angle E. EF is the hypotenuse. The ratio of the sie opposite angle E to the hypotenuse in DEF is The ratios are the same. opposite e. This represents sin B an sin E, because both are the ratio hypotenuse. 2. Ricaro is staning 75 feet away from the base of a builing. The angle of elevation from the groun where Ricaro is staning to the top of the builing is 32. What is, the height of the builing, to the nearest tenth of a foot? sin32 = cos 32 = tan32 = Page 74 of 182 Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents

3 Solution: You want to know the length of the sie opposite the 32 angle, an you know the length of the sie ajacent to the 32 angle. So, use the tangent ratio. Substitute for the opposite sie, 75 for the ajacent sie, an 32 for the angle measure. Then solve. The builing is about 46.9 feet tall. tan32 = 75 75tan32 = An airplane is at an altitue of 5,900 feet. The airplane escens at an angle of 3. About how far will the airplane travel in the air until it reaches the groun? sin3 = cos3 = tan3 = Solution: Use sin3 to fin the istance the airplane will travel until it reaches the groun,. Substitute for the hypotenuse, 5,900 for the opposite sie, an 3 for the angle measure. Then solve. sin3 = 5,900 = 5,900 sin3 5, ,811 The airplane will travel about 1,000 feet until it reaches the groun. Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents Page 75 of 182

4 4. Triangle ABC is a right triangle. What is the best approimation for m C? sin cos tan Solution: Fin the trigonometric ratios for angle C. sin C = cos C = tan C = Using the table, cos , so m C 22.6, or using trigonometric inverses, sin 1 5 = 22.6, cos 1 = 12 5 = 22.6, or tan 1 12 = Page 76 of 182 Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents

5 SAMPLE ITEMS 1. In right triangle ABC, angle A an angle B are complementary angles. The value of cos A is 5. What is the value of sin B? 5 A. B. 12 C. 12 D. 5 Correct Answer: A 2. Triangle ABC is given below. What is the value of cos A? A. 3 5 B. 3 4 C. 4 5 D. 5 3 Correct Answer: A Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents Page 77 of 182

6 3. In right triangle HJK, J is a right angle an tan H = 1. Which statement about triangle HJK must be true? A. sin H = 1 2 B. sin H = 1 C. sin H = cos H D. sin H = 1 cos H Correct Answer: C 4. A 12-foot laer is leaning against a builing at a 75 angle with the groun. Which equation can be use to fin how high the laer reaches up the sie of the builing? A. sin75 = 12 B. tan75 = 12 C. cos75 = 12 D. sin75 = 12 Correct Answer: D Page 78 of 182 Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents

7 5. A hot air balloon is 1,200 feet above the groun. The angle of epression from the basket of the hot air balloon to the base of a monument is 54. Which equation can be use to fin the istance,, in feet, from the basket of the hot air balloon to the base of the monument? A. sin54 = 1200 B. sin54 = 1200 C. cos 54 = 1200 D. cos 54 = 1200 Correct Answer: B Georgia Milestones Geometry EOC Stuy/Resource Guie for Stuents an Parents Page 79 of 182