Name: _________________ Date: _________________ Foundations 20 Unit 3: Actute Triangle Trigonometry Review Assignment Mrs. Boughen

DUE: FRIDAY, MARCH 20 Multiple Choice Identify the choice that best completes the statement or answers the question. ____

1. Solve for the unknown side length. Round your answer to one decimal place.

t 3.5 = sin58° sin47° a. b. c. d. ____

2. Which expression describes the ratios of side-angle pairs in ∆QRS?

a. b. c. d. ____

4.1 4.7 5.6 5.1

q(sin Q) = r(sin R) = s(sin S) q(sin R) = r(sin S) = s(sin Q) q s r = = sinS sinQ sinR q r s = = sinS sinQ sinR

3. In ∆DEF, ∠D = 61°, d = 23.9 cm, and ∠E = 38°. Determine the length of side e to the nearest tenth of a centimetre. a. b. c. d.

16.8 cm 16.0 cm 17.6 cm 18.4 cm

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4. How you would determine the indicated angle measure, if it is possible?

a. b. c. d. ____

5. How you would determine the indicated angle measure, if it is possible?

a. b. c. d. ____

not possible primary trigonometric ratios the cosine law the sine law

6. What information do you need to know about an acute triangle to use the cosine law? a. b. c. d.

____

the cosine law not possible primary trigonometric ratios the sine law

two sides and any angle two angles and any side all the angles all the sides

7. How you would determine the indicated angle measure, if it is possible?

a. b. c. d.

the sine law not possible primary trigonometric ratios the cosine law

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Short Answer: Show all of your work and calculations!!! Make sure to include LET statements and write concluding sentences for all word problems. 8. Determine the length of c to the nearest tenth of a centimetre.

9. Determine the measure of θ to the nearest degree.

10. Determine the measure of α to the nearest degree.

11. In ∆ABC, ∠A = 65°, a = 23.5 cm, and ∠C = 71°. Draw the triangle. Determine the length of side c to the nearest tenth of a centimetre.

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12. In ∆QRS, r = 4.1 cm, s = 2.7 cm, and ∠R = 88°. Draw the triangle. Determine the measure of ∠S to the nearest degree.

. 13. Determine the length of w to the nearest tenth of a centimetre.

14. Determine the measure of θ to the nearest degree.

15. In ∆ABC, a = 108 cm, b = 100 cm, and c = 124 cm. Draw the triangle. Determine the measure of ∠C to the nearest degree.

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16. A kayak leaves a dock on Lake Athabasca, and heads due north for 2.8 km. At the same time, a second kayak travels in a direction N70°E from the dock for 3.0 km. How you can determine the distance between the kayaks?

. 17. How long, to the nearest inch, is the left rafter in the roof shown?

18. A radar operator on a ship discovers a large sunken vessel lying parallel to the ocean surface, 180 m directly below the ship. The length of the vessel is a clue to which wreck has been found. The radar operator measures the angles of depression to the front and back of the sunken vessel to be 52° and 67°. How long, to the nearest tenth of a metre, is the sunken vessel?

. 19. In a parallelogram, two adjacent sides measure 8.4 cm and 7.2 cm. The shorter diagonal is 10.5 cm. Determine, to the nearest degree, the measures of the larger angles in the parallelogram.

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Problem 20. A radio tower is supported by two wires on opposite sides. On the ground, the ends of the wire are 280 m apart. One wire makes a 60° angle with the ground. The other makes a 66° angle with the ground. Draw a diagram of the situation. Then, determine the length of each wire to the nearest metre. Show your work.

. 21. Stella decided to ski to a friend’s cabin. She skied 8.0 km in the direction N40°E. She rested, then skied S30°E and arrived at the cabin. The cabin is 9.5 km from her home, as the crow flies. Determine, to the nearest tenth of a kilometre, the distance she travelled on the second leg of her trip. Show your work.

. 22. Two Jasper National Park rangers in their fire towers spot a fire. Determine the distances, to the nearest tenth of a kilometre, from each tower to the fire. Show your work.

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23. Determine, to the nearest centimetre, the perimeter of the triangle. Show your work.

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ID: A

DUE: FRIDAY, MARCH 20 Answer Section MULTIPLE CHOICE 1. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 3.1 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Side-angle relationships in acute triangles KEY: primary trigonometric ratios 2. ANS: C PTS: 1 DIF: Grade 11 REF: Lesson 3.1 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Side-angle relationships in acute triangles KEY: primary trigonometric ratios 3. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Proving and applying the sine law KEY: sine law 4. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. TOP: Solving problems using acute triangles KEY: sine law| cosine law| primary trigonometric ratios 5. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Solving problems using acute triangles KEY: sine law| cosine law| primary trigonometric ratios 6. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 3.3 OBJ: 3.2 Explain the steps in a given proof of the sine law or cosine law. TOP: Proving and applying the cosine law KEY: cosine law 7. ANS: C PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. | 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Solving problems using acute triangles KEY: sine law| cosine law| primary trigonometric ratios SHORT ANSWER 8. ANS: c = 42.7 cm PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Proving and applying the sine law KEY: sine law 9. ANS: θ = 57° PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Proving and applying the sine law KEY: sine law

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ID: A 10. ANS: α = 69° PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Proving and applying the sine law KEY: sine law 11. ANS: c = 24.5 cm PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Proving and applying the sine law KEY: sine law 12. ANS: ∠S = 41° PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. TOP: Proving and applying the sine law KEY: sine law 13. ANS: w = 27.3 cm PTS: 1 DIF: Grade 11 REF: Lesson 3.3 OBJ: 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. TOP: Proving and applying the cosine law KEY: cosine law 14. ANS: θ = 57° PTS: 1 DIF: Grade 11 REF: Lesson 3.3 OBJ: 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. TOP: Proving and applying the cosine law KEY: cosine law 15. ANS: ∠C = 73° PTS: 1 DIF: Grade 11 REF: Lesson 3.3 OBJ: 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. TOP: Proving and applying the cosine law KEY: cosine law 16. ANS: Since the measures of two sides and a contained angle are given, I would use the cosine law. PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.1 Draw a diagram to represent a problem that involves the cosine law or the sine law. | 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Solving problems using acute triangles KEY: cosine law

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ID: A 17. ANS: 30′4′′ PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Solving problems using acute triangles KEY: sine law 18. ANS: 217.0 m PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.1 Draw a diagram to represent a problem that involves the cosine law or the sine law. | 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Solving problems using acute triangles KEY: sine law| primary trigonometric ratios 19. ANS: 96° PTS: 1 DIF: Grade 11 REF: Lesson 3.4 OBJ: 3.1 Draw a diagram to represent a problem that involves the cosine law or the sine law. | 3.3 Solve a problem involving the cosine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Solving problems using acute triangles KEY: cosine law

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ID: A PROBLEM 20. ANS: Let the x and y be the length of the wires. The third angle is 180° – 66° – 60° = 54°.

Use the sine law to determine the length of each wire: y 280 x 280 = = sin60° sin54° sin66° sin54°

x=

280sin66° sin54°

x=

280sin60° sin54°

= 316.177. . . = 299.730. . . The wires are 316 m and 300 m long. PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.1 Draw a diagram to represent a problem that involves the cosine law or the sine law. | 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Proving and applying the sine law KEY: sine law| primary trigonometric ratios

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ID: A 21. ANS:

Because the lines are parallel, the angle beside the 30° angle is also 40°. The entire angle is 70°. sinz sin70° = 8.0 9.5 ÁÊ sinz ˜ˆ˜ ÁÊ ˜ˆ ˜˜ = 8.0 ÁÁÁ sin70° ˜˜˜ 8.0ÁÁÁÁ ˜ Á ˜ Ë 8.0 ¯ Ë 9.5 ¯

z = sin −1 (0.7913. . . ) z = 52.3090. . . ° x + 70° + z = 180° x + 70° + 52.309...° = 180° x = 57.690...° d 9.5 = sinx sin 70° d 9.5 = sin57.690. . . ° sin 70° d = 8.544. . . Stella travelled 8.5 km. PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.1 Draw a diagram to represent a problem that involves the cosine law or the sine law. | 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Proving and applying the sine law KEY: sine law

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ID: A 22. ANS: Let ∠C represent the measure of the remaining unknown angle. ∠A + ∠B + ∠C = 180° 64° + 48° + ∠C = 180° ∠C = 68° Let b represent the distance from tower A to the fire. b c = sinB sinC

4.2 b = sin48° sin68° ÊÁ 4.2 ˆ˜ ˜˜ b = sin48° ÁÁÁÁ ˜˜ sin68° Ë ¯ b = 3.366. . . The distance from tower A to the fire is 3.4 km. Let a represent the distance from tower B to the fire. a c = sinA sinC 4.2 a = sin64° sin68° ÊÁ 4.2 ˆ˜ ˜˜ a = sin64° ÁÁÁÁ ˜˜ sin68° Ë ¯ a = 4.071. . . The distance from tower B to the fire is 4.1 km. PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Proving and applying the sine law KEY: sine law

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ID: A 23. ANS: Determine the measure of ∠N. ∠L + ∠M + ∠N = 180° 80° + 50° + ∠N = 180° ∠N = 50° ∆LMN is isosceles because ∠M = ∠N. So, m = 14 cm because it is also opposite a 50° angle. Determine the length of l. l m = sinL sinM

14 l = sin80° sin50° ÊÁ 14 ˆ˜ ˜˜ l = sin80° ÁÁÁÁ ˜˜ sin50° Ë ¯ l = 17.998. . . Perimeter = l + m + n Perimeter = 17.998... + 14 + 14 Perimeter = 45.998... The perimeter is 46 cm. PTS: 1 DIF: Grade 11 REF: Lesson 3.2 OBJ: 3.5 Solve a problem involving the sine law that requires the manipulation of a formula. | 3.6 Solve a contextual problem that involves the cosine law or sine law. TOP: Proving and applying the sine law KEY: sine law

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