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Pre-Class Problems 8 for Monday, February 25 Problems which are due at the beginning of class: 1. 2. Use your calculator to approximate the following to four decimal places. (Round to the nearest ten-thousandth.) 28 a. sin ( 105 ) b. sec 19 Find the exact value of , x, and z. Then approximate the value of x and z to the nearest tenth. 29.6 z x 48.3 3. A 25-foot ladder is leaning against the top of a vertical wall. If the top of the ladder makes an angle of 52 with the wall, then find the height of the wall. Find the exact value and then round to the nearest tenth. 4. One website that you used for these pre-class problems other than mine. These are the type of problems that you will be working on in class. These problems are from Lesson 7. You can go to the solution for each problem by clicking on the problem letter. Objective of the following problems: To use a calculator to approximate the value of a trigonometric function of an angle. 1. Use your calculator to approximate the following to four decimal places. (Round to the nearest ten-thousandth.) a. sin ( 215 ) d. 26 cos 11 b. sec 9 7 c. cot 289 e. tan 1890.4 f. csc 14 Objective of the following problems: To solve for unknowns in a given right triangle. To use a calculator to obtain approximations for the exact answers. 2. Solve for the following variables. a. Find the exact value of , x, and y. Then approximate the value of x and y to the nearest hundredth. x y 26.8 b. 34.7 Find the exact value of , x, and z. Then approximate the value of x and z to the nearest tenth. 49.3 z x 24.2 Objective of the following problems: To take a written description and produce a right triangle with known information and one unknown. The unknown information is represented by a variable. Then use a trigonometric function to obtain an equation containing the variable. Solve this equation for the exact value of the variable. Then approximate the exact value of the variable as indicated. For some of these problems, you will need the definition for angle of elevation and for angle of depression. An angle of elevation and an angle of depression are both acute angles measured with respect to the horizontal. An angle of elevation is measured upward and an angle of depression is measured downward. The angle below is an angle of elevation from the point A to the point B above. The angle below is an angle of depression from the point B to the point A below. A B B A 3a. The angle of depression from the top of a building to an object on the ground is 40 . If the object is 85 feet from the base of the building, then find the height of the building. Find the exact value and then round to the nearest tenth. 3b. The angle of depression from the top of a 150-foot building to an object on the ground is 24.7 . How far is the object from the base of the building? Find the exact value and then round to the nearest hundredth. 3c. From a point P on the ground, the angle of elevation to the top of a 60-yard tree is 35 . What is the distance from the point P to the top of the tree? Find the exact value and then round to the nearest hundredth. 3d. The angle of elevation of the string from the ground to a kite is 48.6 . If the length of the string is 125 meters, then how far is the kite above ground? Find the exact value and then round to the nearest tenth. 3e. An observer on the ground is 105 yards from the point directly beneath a balloon. If the angle of elevation from the observer to the balloon is 28 , then how far is the balloon from the observer? Find the exact value and then round to the nearest hundredth. 3f. A ladder is leaning against the top of a vertical wall. The top of the ladder makes an angle of 34 with the wall. If the height of the wall is 6 meters, then find the length of the ladder. Find the exact value and then round to the nearest tenth. 3g. The angle of elevation from an object on the ground to the top of a building is 57 . If the object is 95 meters from the top of the building, then find the distance from the object to the base of the building. Find the exact value and then round to the nearest thousandth. 3h. From a point on the ground which is 40 feet from the base of a tree, the angle of elevation to the top of the tree is 72.3 . What is the height of the tree? Find the exact value and then round to the nearest tenth. 3i. A ladder is leaning against the top of a 15-yard vertical wall. The bottom of the ladder makes an angle of 24.1 with the ground. How far is the bottom of the ladder from the base of the wall? Find the exact value and then round to the nearest hundredth. Additional problems available in the textbook: Evaluating Trigonometric Functions with a Calculator - Parts a and b on Page 157. Examples 7 and 9 on page 158. Page 161 … 63, 64, 65, 66, 67, 70, 71. Page 211 … 5, 6, 7, 8, 19, 20, 21, 22. Examples 1 and 2 on Page 205. Requires a system of equations to solve: Page 162 … 72. Page 211 … 23, 24, 25. For Problem 23, use 47.67 for 47 40 . Examples 3 on Page 206. Solutions: 1 a. sin ( 215 ) Answer: 0.5736 NOTE: In order to find the sine of the angle 215 , the mode of your calculator needs to be set on Degrees. If your calculator is set on Radians, then you would incorrectly give an answer of 0 . 9802 . Since you know that the terminal side of the angle 215 is in the second quadrant, where sine is positive, then you would know that this value is not correct. Back to Problem 1. 1 b. sec 9 7 Answer: 1. 6039 NOTE: The secondary key of COS 1 , which is above the COS key, on your calculator is NOT the secant key. It is the key for the inverse cosine function which we will study in Lesson 9. NOTE: Since your calculator does not have a secant key, you will first need 9 to find the cosine of the angle . Do not round this number, which is 7 0 . 6234898019 . Now, find the multiplicative inverse (reciprocal) of this 1 number using your reciprocal key, which is x or 1 / x , in order to obtain 9 the secant of the angle since secant is the reciprocal of cosine. 7 9 NOTE: In order to find the cosine of the angle , the mode of your 7 calculator needs to be set on Radians. If your calculator is set on Degrees, 9 sec then you would incorrectly give an answer of 1.0025 for . Since 7 9 you know that the terminal side of the angle is in the third quadrant, 7 where secant is negative, then you would know that this value is not correct. If the mode of your calculator was set on Radians and you used the secondary key of COS 1 , then your calculator would give you an error message since 9 the number is greater than one. We will learn in Lesson 9 that you can 7 not take the inverse cosine of numbers greater than one. Back to Problem 1. 1c. cot 289 Answer: 0 . 3443 NOTE: The secondary key of TAN 1 , which is above the TAN key, on your calculator is NOT the cotangent key. It is the key for the inverse tangent function which we will study in Lesson 9. NOTE: Since your calculator does not have a cotangent key, you will first need to find the tangent of the angle 289 . Do not round this number, which is 2 . 904210878 . Now, find the multiplicative inverse (reciprocal) of this 1 number using your reciprocal key, which is x or 1 / x , in order to obtain the cotangent of the angle 289 since cotangent is the reciprocal of tangent. NOTE: In order to find the tangent of the angle 289 , the mode of your calculator needs to be set on Degrees. If your calculator is set on Radians, then you would incorrectly give an answer of 37 . 6927 for cot 289 . If the mode of your calculator was set on Degrees and you used the secondary key of TAN 1 , then you would incorrectly give an answer of 89.8017 cot 289 . Since you know that the terminal side of the angle 289 is in the fourth quadrant, where cotangent is negative, then you would know that this value is not correct. Back to Problem 1. 1d. 26 cos 11 Answer: 0.4154 26 , the mode of your 11 calculator needs to be set on Radians. If your calculator is set on Degrees, then you would incorrectly give an answer of 0.9916. NOTE: In order to find the cosine of the angle Back to Problem 1. 1e. tan 1890.4 Answer: 143. 2371 NOTE: In order to find the tangent of the angle 1890.4 , the mode of your calculator needs to be set on Degrees. If your calculator is set on Radians, then you would incorrectly give an answer of 1.1129 . Back to Problem 1. 1f. csc 14 Answer: 1.0095 NOTE: The secondary key of SIN 1 , which is above the SIN key, on your calculator is NOT the cosecant key. It is the key for the inverse sine function which we will study in Lesson 9. NOTE: Since your calculator does not have a cosecant key, you will first need to find the sine of the angle 14 (radians). Do not round this number, which is 0.9906073557. Now, find the multiplicative inverse (reciprocal) of 1 this number using your reciprocal key, which is x or 1 / x , in order to obtain the cosecant of the angle 14 (radians) since cosecant is the reciprocal of sine. NOTE: In order to find the sine of the angle 14 (radians), the mode of your calculator needs to be set on Radians. If your calculator is set on Degrees, then you would incorrectly give an answer of 4.1336 for csc 14 . If the mode of your calculator was set on Radians and you used the secondary key of SIN 1 , then your calculator would give you an error message since the number 14 is greater than one. We will learn in Lesson 9 that you can not take the inverse sine of numbers greater than one. Back to Problem 1. 2a. x y 26.8 To find : 34.7 26.8 90 63.2 Answer: 63.2 To find x: x sin 26.8 x 34.7 sin 26.8 34.7 Answer: Exact: x 34.7 sin 26.8 Approximate: 15.65 NOTE: sin 26.8 0 . 4508775407 y cos 26.8 y 34.7 cos 26.8 34.7 To find y: Answer: Exact: y 34.7 cos 26.8 Approximate: 30.97 NOTE: cos 26.8 0 .8925858185 Back to Problem 2. 2b. 49.3 z x 24.2 To find : 49.3 90 40.7 Answer: 40.7 To find x: 24.2 x tan 49.3 cot 49.3 x 24.2 cot 49.3 x 24.2 24.2 24.2 tan 49 . 3 24 . 2 x tan 49 . 3 x OR x tan 49.3 Answer: 24.2 x x 24 . 2 cot 49 . 3 Exact: OR tan 49.3 Approximate: 20.8 NOTE: tan 49.3 1.162607256 cot 49.3 0 .8601356946 To find z: 24.2 z sin 49.3 csc 49.3 z 24.2 csc 49.3 z 24.2 24.2 24.2 sin 49 . 3 24 . 2 x sin 49 . 3 x OR z sin 49.3 Answer: 24.2 z z 24 . 2 csc 49 . 3 Exact: OR sin 49.3 Approximate: 31.9 NOTE: sin 49.3 0 . 7581343362 csc 49.3 1.31902745 Back to Problem 2. 3a. Top of Building -------------40 y 40 85 feet Object NOTE: Since the angle of depression is 40 , then the angle of elevation is also 40 . y tan 40 y 85 tan 40 85 Answer: Exact: 85 tan 40 ft Approximate: 71.3 ft Back to Problem 3. 3b. Top of Building -------------24.7 150 feet 24.7 Object x NOTE: Since the angle of depression is 24.7 , then the angle of elevation is also 24.7 . 150 x tan 24.7 cot 24.7 x 150 cot 24.7 x 150 OR 150 150 tan 24.7 150 x tan 24.7 x x tan 24.7 Answer: Exact: 150 cot 24.7 ft OR Approximate: 326.12 ft 150 tan 24.7 ft Back to Problem 3. 3c. Top of Tree z 60 yards P 35 60 z sin 35 csc 35 z 60 csc 35 z 60 OR 60 60 sin 35 60 z sin 35 z z sin 35 Answer: Exact: 60 csc 35 yd OR Approximate: 104.61 yd Back to Problem 3. 3d. Kite 125 meters y 48.6 60 yd sin 35 y sin 48.6 y 125 sin 48.6 125 Answer: Exact: 125 sin 48.6 m Approximate: 93.8 m Back to Problem 3. 3e. Balloon z Observer 28 105 yards 105 z cos 28 sec 28 z 105 sec 28 z 105 OR 105 105 cos 28 105 z cos 28 z z cos 28 Answer: Exact: 105 sec 28 yd OR Approximate: 118.92 yd Back to Problem 3. 105 cos 28 yd 3f. Top of Wall 34 z 6 meters 6 z cos 34 sec 34 z 6 sec 34 z 6 OR 6 6 cos 34 6 z cos 34 z z cos 34 Answer: Exact: 6 sec 34 m OR 6 m cos 34 Approximate: 7.2 m Back to Problem 3. 3g. Top of Building 95 meters 57 x x cos 57 x 95 cos 57 95 Object Answer: Exact: 95 cos 57 m Approximate: 51.741 m Back to Problem 3. 3h. Top of Tree y 72.3 40 feet y tan 72.3 y 40 tan 72.3 40 Answer: Exact: 40 tan 72.3 ft Approximate: 125.3 ft Back to Problem 3. 3i. Top of Wall 15 yards 24.1 x 15 x tan 24.1 cot 24.1 x 15 cot 24.1 x 15 15 15 tan 24 . 1 15 x tan 24 . 1 x OR x tan 24.1 Answer: Exact: 15 cot 24.1 yd OR 15 yd tan 24.1 Approximate: 33.53 m Back to Problem 3. Solution to Problems on the Pre-Exam: 13. Given the triangle below, find x. Set up an equation and solve. (4 pts.) 45 x 28 45 x tan 28 cot 28 x 45 cot 28 x 45 45 45 tan 28 45 x tan 28 x OR x tan 28 45 x x 45 cot 28 Answer: OR tan 28 15. From the top of a building, which is 80 meters tall, the angle of depression to an object on level ground below is 15.7 . How far is the object from the top of the building? Draw a picture and label known information. Indicate any variable you use. Set up an equation and solve. (6 pts.) Top of Building -------------15.7 z 80 meters 15.7 Object NOTE: Since the angle of depression is 15.7 , then the angle of elevation is also 15.7 . 80 z sin 15.7 csc 15.7 z 80 csc 15.7 z 80 OR 80 80 sin 15.7 80 z sin 15.7 z z sin 15.7 Answer: 80 80 csc 15.7 m OR sin 15.7 m