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Name: __________________ Class: 1 Use the definitions to evaluate the six trigonometric functions of . In cases in which a radical occurs in a denominator, rationalize the denominator. Date: _____________ 2 Suppose that ABC is a right triangle with C = 90 . If AC = 5 and BC = 4 , find the following quantities. cos A , sin A , tan A a. cos A = b. cos A = c. cos A = d. cos A = e. cos A = 4 41 , sin A = 41 41 , sin A = 41 41 , sin A = 41 5 41 , sin A = 41 41 , sin A = 41 5 41 , tan A 41 41 , tan A = 41 41 , tan A = 41 4 41 , tan A 41 41 , tan A = 41 = 4 5 4 5 5 = 4 5 5 4 3 Determine whether the equation is correct by evaluating each 2 side. Do not use a calculator. Note: Notation such as sin stands for 2 ( sin ) . 2 sin 60 = cos 30 + sin 30 a. sin cos = = b. sin cos = = c. sin cos = = d. sin cos = = e. sin cos = = 35 37 37 , 35 1 37 1 , 35 35 37 35 , 12 35 37 35 , 12 35 37 12 , 37 , tan cot , tan cot , tan cot , tan cot , tan cot = 12 , csc 37 37 = , sec 12 = 35 , csc 37 = 37 , sec 12 = 12 , csc 37 12 = , sec 35 = 12 , csc 37 37 = , sec 35 = 35 , csc 12 12 = , sec 35 = 1 = 1 a. no b. yes 4 Determine whether the equation is correct by evaluating each 2 = = = = 12 35 35 12 37 12 37 35 = 2 = 1 2 37 35 37 = 12 = side. Do not use a calculator. Note: Notation such as sin stands for . 2 cos 60 = 2 cos 30 + 1 a. no b. yes 5 Determine whether the equation is correct by evaluating each 2 side. Do not use a calculator. Note: Notation such as sin stands for 2 ( sin ) 2 2 2 cos 30 + sin 45 + sin 60 = 1 . a. yes b. no 6 Determine whether the equation is correct by evaluating each side. Do not use a calculator. sin 60 cos 30 a. yes PAGE 1 2 ( sin ) b. no cos 60 sin 30 = 1 2 Name: __________________ Class: 7 Carry out the indicated operation. sin cot a. 1 b. 10 Use the given information to determine the value of remaining five trigonometric functions. (Assume that the angle is acute.) If a radical appears in the denominator, rationalize the denominator. cot sin c. 1 d. tan sin e. 2 + 4 tan a. (tan b. (tan c. (tan d. (tan e. (tan 3)(tan 5)(tan 7)(tan 4)(tan 6)(tan sec 21 + + + + + 7) 5) 3) 7) 4) 9 Factor the expression. 2 3 sec a. (3 sec b. (3 sec c. (3 sec d. (3 sec e. (3 sec + 10 sec cot sin 1)(sec 2)(sec 5)(sec 2)(sec 2)(sec sec 8 + + + + + 3) 3) 5) 7) 4) = c. cos sec = b. cos = d. cos sec = e. cos sec PAGE 2 = 2 3 a. cos 8 Factor the expression. tan Date: _____________ = = 2 5 , tan 3 3 5 , csc 2 5 , tan = 3 3 5 , csc 5 5 , tan = 3 3 5 , csc 5 5 , tan = 3 3 5 , csc 5 5 , tan = 3 3 , csc = 5 = 2 5 , cot 5 3 = 2 = 2 5 , cot 5 3 = 2 = 2 5 , cot 5 = 5 2 = 2 5 , cot 5 = 3 5 = 2 5 , cot 3 3 2 = 5 , 2 = 5 , 2 = 5 , 2 = 5 , 2 = 3 5 , 2 Name: __________________ Class: 11 Use the given information to determine the value of remaining five trigonometric functions. (Assume that the angle is acute.) If a radical appears in the denominator, rationalize the denominator. Date: _____________ 14 Rewrite in terms of sine and cosine, and simplify the expression: 4 sin 2 sin cos B = 24 25 7 , tan B 29 29 , csc B = 29 24 7 7 , tan B b. sin B = 25 25 , csc B = 25 24 7 7 , tan B c. sin B = 25 25 , csc B = 25 7 24 1 , tan B d. sin B = 25 25 , csc B = 25 24 e. sin B = 7 , tan B 25 25 , csc B = 25 24 7 = 24 , cot B = 7 , sec B = 7 24 sin sin b. sin c. = 24 , cot B = 7 , sec B = 7 24 4 8 a. a. sin B = + 8 d. + 2 2 e. 4 sin + 2 4 4 sin 2 + 2 8 15 Rewrite in terms of sine and cosine, and simplify the expression: = 7 , cot B = 24 , sec B = 24 7 = 1 , cot B = 24, sec B = 24 = 7 , cot B = 24 , sec B = 24 7 cos + 1 + 1 cos cos + 1 1 cos c. cot + csc d. 2 cos + 1 a. 1 b. 0 16 Refer to the figure. If find AC . e. sin cos A = 45 and AB = 40 cm , 12 Rewrite in terms of sine and cosine, and simplify the expression: 2 cos C cos C 2 sin C sin C (Hint: Factor the numerator.) a. cos(C) sin(C) b. cos(C) + sin(C) c. d. 2cos(C) e. 2sin(C) 1 cos C sin C 13 Rewrite in terms of sine and cosine, and simplify the expression: cos sec cot a. csc b. sec c. cos d. cot a. AC = 40 cm d. AC = 20 3 cm b. AC = 40 5 cm e. AC = c. AC = 20 7 cm e. sin 17 A ladder 14 ft long leans against a building. The ladder forms and angle of 45 with the ground. How high up the side of the building does the ladder reach? a. 7 2 ft b. PAGE 3 5 cm 2 ft c. 14 ft d. 7 ft e. 14 2 ft Name: __________________ Class: Date: _____________ 18 From a point level with and 1000 ft away from the base of a monument, the angle of elevation to the top of the monument is 29.35 . Determine the height of the monument to the nearest foot. ________ ft 22 From a point on ground level, you measure the angle of elevation to the top of a mountain to be 35 . Then you walk 160 m farther away from the mountain and find that the angle of elevation is now 15 . Find the height of the mountain. a. 77.5 m b. 69 m 19 Find the area of the triangle. c. 112 m d. 43 m e. 75 m 23 In the figure, AB = 16 in . Express x as a function of . area = ________ in. 2 20 Find the area of the triangle. Use a calculator and round your final answer to two decimal places. a. 3 tan b. 16 cos c. 3 cos 16 cos 3 cot 16 tan d. 3 sin e. 16 sin 16 cot 3 tan 24 Use the definitions (not a calculator) to evaluate the six trigonometric functio of the angle. 180 a. 5.91 cm b. 1.04 cm 2 2 c. 2.95 cm d. 2.08 cm 2 e. 11.82 cm 2 2 21 An observer in a lighthouse is 66 ft above the surface of the water. The observer sees a ship and finds the angle of depression to be 0.5 . Estimate the distance of the ship from the base of the lighthouse. a. 7525 ft b. 7545 ft PAGE 4 c. 7585 ft d. 7605 ft e. 7565 ft a. sin 180 csc 180 b. sin 180 csc 180 c. sin 180 csc 180 d. sin 180 csc 180 e. sin 180 csc 180 = 0 cos 180 = 1 is undefined sec 180 = 1 is undefined cos 180 = 1 = 0 sec 180 = 1 = 1 cos 180 = 0 = 1 sec 180 is undefined = 0 cos 180 = 1 is undefined sec 180 = 1 = 0 cos 180 = 1 = 1 sec 180 is undefined tan 180 cot 180 tan 180 cot 180 tan 180 cot 180 tan 180 cot 180 tan 180 cot 180 = 0 is undefined = 0 is undefined is undefined = 0 is undefined = 0 = 0 is undefined Name: __________________ Class: Date: _____________ 25 Use the definitions (not a calculator) to evaluate the six trigonometric functions 26 Use of the the definitions of the trigonometric functions to choose the angle. correct table. 990 a. sin( 990 ) = 0 cos( b. sin( cos( c. sin( cos( d. sin( cos( e. sin( cos( 990 ) = 1 990 ) = 1 990 ) = 0 990 ) = 0 990 ) = 1 990 ) = 1 990 ) = 0 990 ) = 1 990 ) = 0 a. tan( cot( tan( cot( tan( cot( tan( cot( tan( cot( 990 990 990 990 990 990 990 990 990 990 ) is undefined ) = 0 ) is undefined ) = 0 ) = 0 ) is undefined ) is undefined ) = 0 ) = 0 ) is undefined csc( sec( csc( sec( csc( sec( csc( sec( csc( sec( 990 990 990 990 990 990 990 990 990 990 ) = 1 ) is undefined ) is undefined ) = 1 ) is undefined ) = 1 ) = 1 ) is undefined ) = 1 ) is undefined sin cos tan 0 0 1 0 90 1 0 1 180 0 1 0 270 0 1 540 0 1 0 sin cos tan 0 1 0 0 90 0 1 undefined 180 1 0 0 270 0 1 undefined 540 0 1 0 sin cos tan 0 0 1 undefined 90 1 undefined 0 180 0 1 270 540 0 1 b. c. 1 undefined undefined 0 1 0 ..to be continued PAGE 5 Name: __________________ Class: continuation Date: _____________ 29 Evaluate the expression using the concept of a reference angle. d. sin( sin cos tan 0 0 1 0 90 0 undefined 180 0 1 0 270 0 undefined 540 0 1 0 1 1 1 2 a. b. 510 ) 1 5 5 d. 5 c. 1 2 cos 675 = cos( cos tan 0 0 1 0 90 1 0 undefined 180 0 1 0 270 0 undefined 540 0 1 675 ) = sin 675 = sin 1 5 2 30 Evaluate each expression using the concept of a reference angle. e. e. sin( 675 ) = 31 Evaluate the expression using the concept of a reference angle. cot( 600 ) a. 3 5 0 b. c. 3 3 d. 5 5 e. 1 5 3 3 27 Evaluate the expression using the concept of a reference angle. 32 Evaluate the expression using the concept of a reference angle. cos 135 tan 510 1 a. 7 b. 7 2 2 2 c. 2 2 d. 2 7 e. a. b. 2 3 3 3 2 d. 2 c. 1 2 e. 3 3 28 Evaluate the expression using the concept of a reference angle. cos( 1 6 6 b. 6 a. PAGE 6 33 Use the given information to determine the area of the triangle. 480 ) c. d. 1 2 e. 1 2 6 2 Two of the sides are 3 m and 6 m, and the included angle is 150 .