Day 4 Trig Applications HOMEWORK 1. In ΔABC, a = 0, b = 1, and mc = 44º a) Find the length of side c to the nearest integer. b) Find the area of ΔABC to the nearest tenth.. In ΔABC, ma = 50º, a = 40, b = 45, and B is obtuse. Find mc to the nearest degree. 3. How many distinct triangles can be constructed if ma = 30º, a = 8, and b = 10? 4. Two forces of 1 pounds and 0 pounds act on a body with an angle of 60º between them. Find the magnitude of the resultant to the nearest pound. Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 1
5. A ship at sea heads directly toward a cliff on the shoreline. The accompanying diagram shows the top of the cliff, D, sighted from two locations, A and B, separated by distance, S. If mdac = 30, mdbc = 45, and S = 10 feet, what is the height of the cliff, to the nearest foot? 6. Two forces act on a body, making angles of 15º and 37º with the resultant. If the larger force is 50 pounds, what is the magnitude of the resultant to the nearest pound? 7. Gregory wants to build a garden in the shape of an isosceles triangle with one of the congruent sides equal to 1 yards. The area of his garden will be 55 square yards. a) Find the number of possible triangles. b) Find, to the nearest degree, the three angles of the triangle(s). 8. In the accompanying diagram of ΔABC, ma = 65, mb = 70, and the side opposite vertex B is 7. Find the length of the side opposite vertex A, and find the area of ΔABC, to the nearest hundredth. Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 13
Day 6 HOMEWORK Express in terms of a constant or a single trig function. 1) csc x cos x tan x ) cot sec 3) tan x sin x 4) cos sec 5) tan 1 cot 1 6) csc x sec x Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 0
Prove the following identities 7) csc x 1 1 cot cot 8) cos x 1 cot 9) 3 sin x sin xcos x sin x 10) cot x(1 sin x) cot x cos x Choose the appropriate equivalent expression. 1 11. sin sin is equivalent to sin 1) cos θ ) cos θ 3) 1 cos θ 4) 1 + cos θ (1 cos A) 1. sin A cos Acos A 1) 1 ) 3) is equivalent to sin A 4) cos A Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 1
Day 8 HOMEWORK Solve each equation for θ in the interval 0 θ 360. Round to nearest degree. 1. 6cos 5sin 7. cos7 4sec Solve each equation for θ in the interval 0 θ 360. Round answers to nearest 10 minutes. 3. sec 3sec9 0 4. 4sin sin 3 Solve each equation for θ in the interval 0 θ 360. Round answers to nearest minute. 5. tan 4tan 0 6. sin csc 3 Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 7
Homework 11 1 1 cosa sin A 1 1 cosa cos A 1 1 cosa tan A 1 cos A 1. 4 cos a, 0 < a < 5 Find the exact values of: a) sin a b) cos a c) tan a. Rewrite the expression without double or half angles, given that 0 Then simplify the expression. cos sin cos Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 36
A. Trig Applications Day 1 Unit 10 REVIEW Law of Sines: a b c Law of Cosines: sin A sin B sin C c a b abcosc Area of a Triangle: Area 1 sin ab C How many triangles? Use Law of Sines 1. In ABC, a 1, 1 sin A, and 5 1 sin B. Find b. 4. ABC has ma30, mb 45, and a 6. Find the exact value of b. 3. Find the number of triangles that satisfy the given conditions. You need not solve any triangle. a) mb 50, a 9, b 5 b) ma6, c 1, and a 1 c) ma68, b 3, and a 30 d) mc 1, b 6, and c 4 e) mb 44, a 16, and b 17 Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 37
4. In ABC, ma57, a 1, and b 9. Find the measure of angle B to the nearest degree. 5. In ABC, ma50, AC 1, and AB 8. Find the area of the triangle to the nearest tenth. 6. In ABC, a 6, b 8, c 9. Find cos B to four decimal places. 7. In ABC, a 5, b 8, c 9. Find mbto the nearest degree. 8. In ABC, mc 5, a 6, and b 7. Find c to the nearest integer. 9. In ABC, ma100, b 9, and c 6. Find a to the nearest integer. 10. In ABC, ma 45, b 4, and c 5. Find the area of ABC to the nearest integer. 11. If the area of ABC is 4, with a 1 and mc 150, find b. Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 38
1. Forces F1 and F have magnitudes of 40 N and 35 N, respectively, and act upon a body at an angle of 60 to each other. Find the magnitude and direction, to the nearest whole number, of the resultant of these two forces. 13. A 48 N force and another force act upon a body to produce a resultant force that has a magnitude of 70 N. The 48 N force makes an angle of 6 with the resultant. Find the magnitude of the second force to the nearest Newton. 14. In ABC, a 19, c 10, and ma = 111. Which statement can be used to find the value of C? (1) sin C 10 19 19 sin 69 () sin C 10 10sin1 (3) sin C 19 10sin 69 (4) sin C 19 15. Two forces of 40 pounds and 0 pounds, respectively, act simultaneously on an object. The angle between the two forces is 40. Find the magnitude of the resultant, to the nearest tenth of a pound. Find the measure of the angle, to the nearest degree, between the resultant and the larger force. Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 39
16. Sam is designing a triangular piece for a metal sculpture. He tells Martha that two of the sides of the piece are 40 inches and 15 inches, and the angle opposite the 40-inch side measures 10. Martha decides to sketch the piece that Sam described. How many different triangles can she sketch that match Sam s description? (1) 1 () (3) 3 (4) 0 17. A ship at sea is 70 miles from one radio transmitter and 130 miles from another. The angle between the signals sent to the ship by the transmitters is 117.4. Find the distance between the two transmitters, to the nearest mile. 1 18. In ABC, AC 18, BC 10, and cosc. Find the area of ABC to the nearest tenth. 19. In ABC, if AC 1, BC 11, and ma 30, angle C could be (1) an obtuse angle, only () an acute angle, only (3) a right angle, only (4) either an obtuse angle or an acute angle 0. In ABC, m A 53, m B 14, and a = 10. Find b to the nearest integer. Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 40
1. A surveyor is mapping a triangular plot of land. He measures two of the sides and the angle formed by these two sides and finds that the lengths are 400 yards and 00 yards and the included angle is 50. What is the measure of the third side of the plot of land, to the nearest yard? What is the area of this plot of land, to the nearest square yard?. One force of 0 pounds and one force of 15 pounds act on a body at the same point so that the resultant force is 19 pounds. Find, to the nearest degree, the angle between the two original forces. 3. A ship at sea heads directly toward a cliff on the shoreline. The accompanying diagram shows the top of the cliff, D, sighted from two locations, A and B, separated by distance S. If mdac 30, mdbc 45 and S = 30 feet, what is the height of the cliff, to the nearest foot? Answers to Trig Apps Review Packet: 1. 15 13. 64 N. 6 14. 4 3. a) 0 b) 1 c) d) 0 e) 1 15. y 13, 56.8 lbs. 4. 39 16. 1 5. 36.8 cm 17. 174 mi 6. 0.4907 18. 77.9 7. mb 6 19. 4 8. c = 6 0. b = 3 9. a = 1 1. side = 31 yd, area = 30,64 yd 10. 7. 116 11. 14 3. 41 ft 1. 65 N at an angle of 3 to the 35 N force OR 8 to the 40 N force Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 41
B. Verifying and Proving Trig Identities 1. Express in terms of a constant or a single function of. a) 1 sin b) tan 1 cot 1. Prove the following: sin a) cot sin b) sin csc csc c) cot x csc x cos x d) 1 cos cos cot sin sin Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 4
C. Solving Trig Equations Solve each equation for θ in the interval 0 θ 360. It may be necessary to approximate answers to the nearest degree. 1. (sin θ + 1)(tan θ 1) = 0. sin 1 csc Find θ to the nearest degree if 0 θ 360 (**Remember if you can t factor, use quadratic formula) 3. 3 csc θ + csc θ = 1 4. cos θ sin θ = 1 5. sec θ = sec θ + 6. sec θ = tan θ + 3 7. 3 cos θ 5 cos θ 4 = 0 8. 4 sin θ sin θ 3 = 0 9. cos1sin 10. cos sin Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 43
D. Sum and Difference Find the exact value in simplest form for each of the following. 1. cos 105. cos 75 3. sin 165 4. tan 55 If x is in quadrant I, sin x = 1 13, y is in quadrant II, and cos y = 4, find the value of each of the following: 5 5. cos (x - y) 6. sin (90 + x) 7. sin (x y) 8. tan (x - y) Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 44
E. Half Angle & Double Angle 1. 7 If sin, and 70 360, find 5 a. cos b. sin. 3 If cos,and 0 90, find 5 a. sin b. tan 3. 4 If cos,and 90 180, find 5 a. tan b. cos 4. 8 If tan, and 0 90, find 15 a. sin 5. 3 91 If cot,and 70 360, find 91 a. tan b. sin b. cos Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 45
A. Trig Applications Unit 10 Review Answers 1. 15. 6 3. a) 0 b) 1 c) d) 0 e) 1 4. 39 5. 36.8 cm 6. 0.4907 7. mb 6 8. c = 6 9. a = 1 10. 7 11. 14 1. 165 N at an angle of 3 to the 35 N force OR 8 to the 40 N force 13. 64 N 14. 4 15. y 13, 56.8 lbs. 16. 1 17. 174 mi 18. 77.9 19. 4 0. b = 3 1. side = 31 yd, area = 30,64 yd. 116 3. 41 ft B. Verifying and Proving Trig Identities All proofs verify 1a) cos 1b) tan C. Solving Trig Equations 1) 45 0, 10 0, 5 0, 330 0 ) 30 0, 150 0, 70 0 3) 70 0 4) 0 0, 360 0 5) 60 0, 180 0, 300 0 6) 63 0, 135 0, 43 0, 315 0 7) 16 0, 34 0 8) 1 0, 319 0 9) 49 0, 131 0, 70 0 10) 90 0, 70 0 D. Sum and Difference 1) 5) 6 4 16 65 ) 6) E. Half Angle & Double Angle 1a) 7 1b) 10 3a) 3 3b) 5a) 91 5b) 13 6 4 5 13 336 65 a) 7 5 4a) 41 50 3) 7) 5 5 17 7 6 4 63 8) 65 b) 4b) 4 7 40 89 4) 3 63 16 Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 46
Trig Reference Page Things you NEED to know 180 radians deg rees deg rees radians 180 s r or angle = arc radius s = r sin (x,y) (cos, sin ) tan = or cos y x Area of a Triangle 1 k ab sin C Area of a parallelogram A= ab sin C Functions of the Sum of Two Angles sin (A + B) = sin A cos B + cos A sin B cos (A + B) = cos A cos B sin A sin B tan A tan B tan( AB) 1 tanatanb Functions of the Difference ot Two Angles sin (A - B) = sin A cos B - cos A sin B cos (A - B) = cos A cos B + sin A sin B tan A tan B tan( AB) 1 tanatanb Law of Sines a b c sin A sin B sinc Functions of the Double Angle sin A = sin A cos A Law of Cosines a = b + c bc cos A cos A = cos A sin A cos A = cos A -1 tana tan A cos A = 1 sin A 1 tan A Functions of the Half Angle 1 1 cosa sin A Pythagorean Identities sin x cos x 1 1 1 cosa cos A 1 tan x sec x 1 1 cosa tan A 1 cos A 1cot x csc x Unit 10: Trig Apps and Identities Notetaking Guide3/6/01 47