Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
page 1 Answers Problem Set 6 Math 141 Use the button on your calculator instead of typing in 3:14: Present exact values (preferred) or round approximations to three decimal places. 1. Use the unit square to …nd the exact values for each of the following p 2 1 (a) sin 45 = p = 2 2 (b) tan 45 = 1 (c) sin2 45 + cos2 45 = 1 2. The radius of a circle is 8 m. 8 m = 2: 793 m 9 Solution: The circumference of the circle is C = 2 r = 2 (8 m) = 16 m. This is the arc belonging to the central angle of 360 . The rest is just a ratio problem. angle arc 360 C 20 x (a) Find the length of an arc subtended by a central angle of 20 . x 20 = C 360 =) x=C 20 360 = C 16 m 8 = = 18 18 9 m ' 2: 793 m 32 m2 = 11: 170 m2 9 Solution: The area of the circle is A = r2 = (8 m)2 = 64 m2 . This is the area of the sector belonging to the central angle of 360 . The rest is just a ratio problem. angle area of sector 360 A 20 x (b) Find the area of a sector subtended by a central angle of 20 . x 20 = A 360 =) x=A 3. Find the polar coordinates for (20; 15). 20 360 = A 64 m2 32 = = 18 18 9 m2 ' 11: 170 m2 (25; 0:644 rad) = (25; 36:870 ) 4. Find the rectangular coordinates for (3; 41 ). (2: 264 ; 1: 968 ) 5. Convert 15 to radians. Round your answer to three decimal places. 0:262 6. Convert 1 radian to degrees. 57: 296 7. Let ABC be a right triangle with sides a; b; and c: (c is the hypotenuse.) Express sin2 in terms of a; b; and c. 1 + cos2 2 8. Find the angle that the straight line y = x + 5 forms with the positive half of the x-axis. 3 33: 690 = 0:588 rad c copyright Hidegkuti, Powell, 2008 page 2 Answers Problem Set 6 Math 141 9. Find the exact value for all six trigonometric function of , based on the following picture. sin = 12 ; 13 cos = 5 ; 12 tan = 12 ; 5 csc = 13 ; 12 sec = 12 ; 5 cot = 5 12 10. Find the radius of a circle if we know that a central angle of 70 subtends an arc of 12 in. 216 in = 9: 822 in 7 11. We are driving toward a tower. The angle of elevation is 28 . Then we drive 550 ft toward the tower. Now the angle of elevation is 40 . How tall is the tower? 798:290 ft Solution: Based on the picture below, we can write a system of linear equations. tan 28 tan 40 H x + 550 H = x = In this system, there are two unknowns, H and x. We will use substitution to solve the system. It is a bit unusual, but common in trigonometry to symbolically carry the trigonometric expressions and only to substitute approximate values into them in the last line. tan 28 tan 40 H =) x + 550 H = =) x = tan 28 (x + 550) x tan 28 + 550 tan 28 550 tan 28 550 tan 28 550 tan 28 tan 40 tan 28 H = tan 40 x = tan 40 c copyright Hidegkuti, Powell, 2008 550 tan 28 tan 40 tan 28 = = = = H = tan 28 (x + 550) H = tan 40 x tan 40 x x tan 40 x tan 40 x (tan 40 x tan 28 tan 28 ) = x = 550 tan 40 tan 28 = 798: 289 78 ( ft) tan 40 tan 28 Problem Set 6 Math 141 12. Find the exact value of 2 sin 4 cos 4 page 3 Answers = 1 13. Green Bay, WI and Mobile, AL have approximately the same longitude. The radius of the earth is approximately 3960 miles. The latitude of Green Bay is 44:5 and that of Mobile is 30: 7 : Find the distance to the nearest mile between the two cities. 303: 6 mi = 954 mi Solution: A circle has radius 3960 miles. Find the length of the arc subtended by a central angle of 44:5 30:7 = 13: 8 . 14. The picture below shows a straight pyramid with a square base. The sides of the base are 10 in long. The other sides are 15 in long. p (a) Find the height of a triangluar edge. 10 2 in = 14: 142 in p (b) Use part a) to …nd the height of the pyramid. 5 7 in = 13: 229 in 15. Find the side of a regular polygon with 10 sides that is written in a circle of radius 28. 17: 305 units 16. An object is traveling on a circular orbit of radius 4 meters. It completes 10 cycles in one minute. 4 m m Find the speed of the object. Express your answer in meter per second. = 4: 189 3 s s c copyright Hidegkuti, Powell, 2008