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Name ________________________________________ Date __________________ Class__________________ Reteach LESSON 10-1 Right-Angle Trigonometry A trigonometric ratio compares the lengths of two sides of a right triangle. The values of the ratios depend upon one of the acute angles of the triangle, denoted by θ. Use SOHCAHTOA to remember the relationships between the sides of a right Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, triangle that correspond to the trigonometric Tangent is Opposite over Adjacent. ratios sine, cosine, and tangent. a Opposite = Hypotenuse c sin θ = cos θ = b Adjacent = Hypotenuse c tan θ = Opposite a = Adjacent b Use the definitions of each ratio and the corresponding values from a given right triangle to find the values of the trigonometric functions for θ. Opposite 14 7 = = Hypotenuse 50 25 sin θ = cos θ = Adjacent 48 24 = = Hypotenuse 50 25 tan θ = Opposite 14 7 = = Adjacent 48 24 Find the value of the sine, cosine, and tangent functions for θ. 1. 2. 3. Opposite = ____________ Hypotenuse sinθ = ____________ sinθ = ____________ cos θ = Adjacent = ____________ Hypotenuse cosθ = ____________ cosθ = ____________ tan θ = Opposite = ____________ Adjacent tanθ = ____________ tanθ = ____________ sin θ = Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 10-6 Holt McDougal Algebra 2 Name ________________________________________ Date __________________ Class__________________ LESSON 10-1 Reteach Right-Angle Trigonometry (continued) The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. The cosecant function (csc θ ) is the reciprocal of the sine function. csc θ = 1 Hypotenuse c = = sin θ Opposite a The secant function (sec θ ) is the reciprocal of the cosine function. sec θ = 1 Hypotenuse c = = cos θ Adjacent b The cotangent function (cot θ ) is the reciprocal of the tangent function. cot θ = 1 Adjacent b = = tan θ Opposite a Use the reciprocal relationship of the ratios to find the values of the reciprocal trigonometric functions. sin θ = 3 5 csc θ = 1 5 = sin θ 3 cos θ = 4 5 sec θ = 1 5 = cos θ 4 tan θ = 3 4 cot θ = 1 4 = tan θ 3 Find the values of the six trigonometric functions for θ. 4. 5. sin θ = ___________ csc θ = 1 = ________________ sin θ cos θ = ___________ sec θ = 1 = ________________ cos θ tan θ = ___________ cot θ = 1 = _________________ tan θ ______________ __________________ ______________ __________________ ______________ __________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 10-7 Holt McDougal Algebra 2 10-1 RIGHT-ANGLE TRIGONOMETRY 10. 47 ft 11. 162 ft 12. 4817 m Practice A 1. a. sinθ = 48 24 = 50 25 b. cos θ = 14 7 = 50 25 Practice C 48 24 = c. tanθ = 14 7 2. 3 4 3 , , 5 5 4 4. 12 5 12 , , 13 13 5 1. 7 24 7 ; ; 25 25 24 2. 3. 60 11 60 ; ; 61 61 11 4. 20 3 5. 3.4 3 3. 9 40 9 , , 41 41 40 b. cos 45° = 7 24 ; cos θ = ; 25 25 tanθ = 7 25 ; csc θ = ; 24 7 sec θ = 25 24 ; cot θ = 24 7 c. cos 45° = x 12 2 2 2 8. sinθ = 12 5 ; cos θ = ; 13 13 tan θ = 12 13 ; csc θ = ; 5 12 sec θ = 13 5 ; cot θ = 5 12 d. x = 12 6. 10 7. 8 8. 9 9. 45 ft 6. 16 7. sinθ = 5.a. Cosine 9. sinθ = Practice B 1. 4 3 4 ; ; 5 5 3 2. 9 40 9 ; ; 41 41 40 3. 12 5 12 ; ; 13 13 5 4. 6 3 5. 44 3 3 6. 7 12 5 12 ; cos θ = ; tan θ = 7. sin θ = 13 13 5 4 3 4 ; ; 5 5 3 10 15 ; cos θ = ; 5 5 tan θ = 6 10 ; csc θ = ; 3 2 sec θ = 15 6 ; cot θ = 3 2 10. 83.5 ft 11. 287 ft 12. 2350 m Reteach 13 13 5 ; sec θ = ; cot θ = 12 5 12 1. 3 5 4 5 3 4 8. sin θ = 3 4 3 ; cos θ = ; tan θ = 5 5 4 2. 12 13 5 13 12 5 csc θ = 5 5 4 ; sec θ = ; cot θ = 3 4 3 3. 8 17 15 17 8 15 csc θ = 9. sin θ = 9 40 9 ; cos θ = ; tan θ = 41 41 40 csc θ = 41 41 40 ; sec θ = ; cot θ = 9 40 9 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A30 Holt McDougal Algebra 2 4. 15 17 17 15 2. a. 12 8 17 17 8 c. 15 15 8 8 15 5 5. sin θ = 13 b. 9 13 csc θ = 5 5 12 cot θ = 12 5 Challenge 1. 220 m 2. 454 m 3. 234 m 4. 70 ft 9 3 or 15 5 e. 12 4 or 9 3 3. An isosceles triangle has 2 sides that are equal in length. Since the hypotenuse is the longest side of a right triangle, the equal sides must be the 2 legs. Therefore, the 2 angles are the same. Since the angles are equal, both 45°, they have the same sine. 12 13 cos θ = sec θ = 13 12 tan θ = d. 10-2 ANGLES OF ROTATION Practice A 5. 54 ft 1. Problem Solving 1. a. a b. tan 75° = a 180 c. 672 yd 2. a. 380 yd b. cos 60° = 380 c c. 760 yd 3. B 4. H 5. D 6. J 2. Reading Strategy 1. a. 5 b. 8 c. 9.95 d. 8 9.95 e. 5 8 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A31 Holt McDougal Algebra 2