Math 130 Unit 3 Practice Questions

Here is a comprehensive set of practice questions covering all 33 objectives for Modules 10 through 13 in Unit 3. These questions are correlated with the provided past assessments (Test 3, Quiz 9, Quiz 10, Quiz 11, Quiz 12, and Quiz 13).

Module 10: Sections 5.1, 5.2

Section 5.1 (5 Topics)

1. Sketching an angle with absolute value less than 360 degrees in standard position

• Q1: Sketch the angle 150° in standard position.
• Q2: Sketch the angle -85° in standard position.
• Q3: Sketch the angle 210° in standard position.

2. Sketching an angle in standard position given in degrees and finding a coterminal angle

• Q1: Sketch -100° in standard position and find one positive coterminal angle.
• Q2: Sketch 280° in standard position and find one negative coterminal angle.
• Q3: Sketch 45° in standard position and find one negative coterminal angle.

3. Sketching an angle in standard position given in radians and finding a coterminal angle

• Q1: Sketch 4π/3 in standard position and find a positive coterminal angle.
• Q2: Sketch -7π/6 in standard position and find a positive coterminal angle.
• Q3: Sketch π/4 in standard position and find a negative coterminal angle.

4. Sketching an angle with absolute value greater than 360 degrees in standard position

• Q1: Sketch the angle 420° in standard position.
• Q2: Sketch the angle -400° in standard position.
• Q3: Sketch the angle 13π/3 in standard position.

5. Arc length and central angle measure

• Q1: A circle has a radius of 4 feet. Find the exact arc length intercepted by a central angle of π/3 radians.
• Q2: Find the distance a wheel of diameter 3 ft travels in one complete rotation.
• Q3: A central angle of 120° intercepts an arc of 15 inches. Find the radius of the circle.

Section 5.2 (8 Topics)

6. Using a calculator to approximate sine, cosine, and tangent values

• Q1: Use a calculator to approximate sin(42°) to four decimal places.
• Q2: Use a calculator to approximate cos(π/7) to four decimal places.
• Q3: Use a calculator to approximate tan(195°) to four decimal places.

7. Sine, cosine, and tangent ratios: Numbers for side lengths

• Q1: Given a right triangle with legs 3 and 4, and hypotenuse 5, find sin(θ) where θ is opposite the side of length 3.
• Q2: In a right triangle where the adjacent side to θ is 8 and the hypotenuse is 17, find cos(θ).
• Q3: For a right triangle with legs 5 and 12, find tan(θ) for the angle adjacent to the side of length 12.

8. Sine, cosine, and tangent ratios: Variables for side lengths

• Q1: A right triangle has legs x and y, and hypotenuse z. Write the ratio for sin(θ) if θ is opposite x.
• Q2: In a right triangle with hypotenuse c and legs a and b, what is tan(θ) for the angle adjacent to a?
• Q3: If a right triangle has an adjacent leg m and opposite leg n, express cos(θ) in terms of m and n.

9. Using the Pythagorean Theorem to find a sine, cosine, or tangent ratio in a right triangle

• Q1: If sin(θ) = 3/5 in a right triangle, find cos(θ).
• Q2: If cos(θ) = 7/25 in a right triangle, find tan(θ).
• Q3: Given tan(θ) = 8/15, use the Pythagorean theorem to find sin(θ).

10. Using the Pythagorean Theorem to find several trigonometric ratios in a right triangle

• Q1: A right triangle has legs 6 and 8. Find all six trigonometric ratios for the angle opposite the side of length 6.
• Q2: If the hypotenuse is 13 and one leg is 5, find csc(θ) and sec(θ) for the angle adjacent to the leg of length 5.
• Q3: Given sin(θ) = 9/41, find the exact values of the remaining five trigonometric functions.

11. Using a trigonometric ratio to find a side length in a right triangle

• Q1: In a right triangle with an angle of 30° and a hypotenuse of 20, find the length of the opposite leg.
• Q2: Find the adjacent leg of a right triangle if the angle is 45° and the hypotenuse is 10.
• Q3: If an angle is 60° and the opposite leg is 15√3, find the hypotenuse.

12. Using trigonometry to find a length in a word problem with one right triangle

• Q1: A ladder leans against a building making an angle of 70° with the ground. If the base of the ladder is 4 ft from the wall, how long is the ladder?
• Q2: A kite is flying on 150 feet of string. If the angle of elevation is 40°, how high is the kite?
• Q3: A tree casts a 20-foot shadow when the sun's angle of elevation is 55°. Find the height of the tree.

13. Using trigonometry to find lengths in a figure involving two right triangles

• Q1: Two buildings are 50 ft apart. From the top of the shorter building, the angle of elevation to the taller is 30° and the angle of depression to its base is 45°. Find the height of the taller building.
• Q2: A surveyor is 100 ft from a tower. The angle of elevation to the top of a given antenna on the tower is 60°, and to the bottom of the antenna is 45°. Find the antenna's length.
• Q3: From a boat, the angle of elevation to the top of a cliff is 25°. After sailing 200 ft closer, the angle is 40°. How high is the cliff?

Module 11: Sections 5.3, 5.7

Section 5.3 (8 Topics)

14. Sketching an angle with absolute value less than 360 degrees, and also its reference angle

• Q1: Sketch 210° and mark its reference angle θ′.
• Q2: Sketch -45° and identify its reference angle θ′.
• Q3: Sketch 300° and identify its reference angle θ′.

15. Reference angles in degrees: Problem type 1

• Q1: Find the reference angle for 135°.
• Q2: Find the reference angle for 250°.
• Q3: Find the reference angle for -70°.

16. Sketching an angle with absolute value greater than 360 degrees, and also its reference angle

• Q1: Sketch 400° and identify its reference angle.
• Q2: Sketch -500° and identify its reference angle.
• Q3: Sketch 750° and identify its reference angle.

17. Reference angles in degrees: Problem type 2

• Q1: Find the reference angle for 560°.
• Q2: Find the reference angle for -800°.
• Q3: Find the reference angle for 1020°.

18. Sketching an angle with absolute value less than 2π radians, and also its reference angle

• Q1: Sketch 5π/6 and mark its reference angle.
• Q2: Sketch -π/3 and mark its reference angle.
• Q3: Sketch 7π/4 and mark its reference angle.

19. Reference angles in radians: Problem type 1

• Q1: Find the reference angle for 11π/6.
• Q2: Find the reference angle for -3π/4.
• Q3: Find the reference angle for 2π/3.

20. Determining the location of a terminal point given the signs of trigonometric values

• Q1: Identify the quadrant in which θ lies if sin(θ) < 0 and cos(θ) > 0.
• Q2: Identify the quadrant in which θ lies if tan(θ) < 0 and sec(θ) < 0.
• Q3: Identify the quadrant in which θ lies if csc(θ) > 0 and cot(θ) < 0.

21. Finding values of trigonometric functions given information about an angle: Problem type 1

• Q1: The point (-4, 3) is on the terminal side of an angle θ. Find sin(θ) and cos(θ).
• Q2: The point (5, -12) is on the terminal side of an angle θ. Find all six trigonometric functions.
• Q3: The point (-2, -5) is on the terminal side of an angle θ. Find tan(θ).

Module 12: Section 7.1

Section 7.1 (7 Topics)

22. Using a trigonometric ratio to find a side length in a right triangle

• Q1: Solve for leg a in a right triangle where A = 35° and hypotenuse c = 14.
• Q2: Solve for leg b in a right triangle where B = 50° and hypotenuse c = 22.
• Q3: Solve for leg a in a right triangle where B = 25° and leg b = 10.

23. Using trigonometry to find a length in a word problem with one right triangle

• Q1: A ship travels 80 miles on a bearing of 120°. How far east has it traveled?
• Q2: A plane flies 150 miles on a bearing of 210°. How far south is it from the starting point?
• Q3: A hiker walks 5 miles east and 12 miles north. What is the direct distance from the start?

24. Using trigonometric functions and the formula d=rt in a real-world situation

• Q1: An airplane flies at 400 mph on a bearing of S30°W for 2 hours. How far south has it traveled?
• Q2: A submarine descends at an angle of 5° at 15 mph. How deep is it after 30 minutes?
• Q3: A plane climbs at 12° at 300 mph. How long to reach an altitude of 3 miles?

25. Using a trigonometric ratio to find an angle measure in a right triangle

• Q1: In a right triangle, leg a = 15 and leg b = 20. Find angle A.
• Q2: In a right triangle, hypotenuse c = 50 and leg a = 30. Find angle B.
• Q3: In a right triangle, leg b = 7 and hypotenuse c = 14. Find angle A.

26. Using trigonometry to find angles of elevation or depression in a word problem

• Q1: A 50-foot building casts a 40-foot shadow. What is the angle of elevation of the sun?
• Q2: From a 100-foot cliff, the angle of depression to a boat is 15°. How far is the boat from the base of the cliff?
• Q3: An airplane climbs 1000 feet over a ground distance of 5000 feet. What is the angle of climb?

27. Solving a right triangle

• Q1: Solve the right triangle where A = 42° and c = 25.
• Q2: Solve the right triangle where a = 18 and b = 24.
• Q3: Solve the right triangle where B = 65° and a = 10.

28. Using trigonometry to find a length in a word problem with two right triangles

• Q1: From a window 40 ft above ground, the angle of elevation to the top of a neighboring building is 20° and the angle of depression to the base is 35°. Find the building's height.
• Q2: A person is standing between two trees. The angle of elevation to the top of the 30 ft tree is 45° and to the top of the 40 ft tree is 60°. How far apart are the trees?
• Q3: From the top of a 150 ft lighthouse, the angles of depression to two boats in a line are 25° and 40°. Find the distance between the boats.

Module 13: Section 8.4

Section 8.4 (4 Topics)

29. Writing a vector in component form given its initial and terminal points

• Q1: Find the component form of the vector with initial point (-2, 3) and terminal point (4, 1).
• Q2: Find the component form of the vector with initial point (0, -5) and terminal point (6, -2).
• Q3: Find the component form of the vector with initial point (1, 1) and terminal point (-3, -4).

30. Vector addition and scalar multiplication: Component form

• Q1: Given u = ⟨3, -2⟩ and v = ⟨1, 5⟩, find u + v.
• Q2: Given u = ⟨15, 20⟩ and v = ⟨-5, 12⟩, find the magnitude of u + v.
• Q3: Given v = ⟨4, -7⟩, find 3v.

31. Finding the magnitude and direction of a vector given its graph

• Q1: A vector extends from (0,0) to (3,4). Find its magnitude and direction angle.
• Q2: A vector extends from (-1,4) to (2,2). Find its magnitude.
• Q3: A vector extends from (-3,-3) to (0,0). Find its direction angle.

32. Finding the direction angle of a vector given in ai+bj form

• Q1: Find the direction angle of v = 5i - 5j.
• Q2: Find the direction angle of v = -2i + 2√3j.
• Q3: Find the exact component form of a vector with magnitude 10 and direction angle 150°.

Chapter 8 Supplementary Topics (1 Topic)

33. Finding the components of a vector given its graph

• Q1: Given a vector drawn from (2, 1) to (5, 6) on a grid, determine its x and y components.
• Q2: Read the graph of vector r pointing left 4 units and up 2 units. Write its component form.
• Q3: From a graph showing u and v added head-to-tail, determine the component form of the resultant r.