Stewart Precalc 6e Section 6.6: The Law of Cosines

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Stewart Precalc 6e Section 6.6: The Law of Cosines 0/57

1 Concept - Law of Cosines formula · Level 1

For triangle ABC with sides \(a\), \(b\), and \(c\) the Law of Cosines states \(c^2 = \) ______.

2 Concept - When to use Law of Cosines · Level 1

In which of the following cases must the Law of Cosines be used to solve a triangle? ASA, SSS, SAS, SSA

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3 Skill - Law of Cosines from figure · Level 2

Use the Law of Cosines to determine the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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4 Skill - Law of Cosines from figure · Level 2

Use the Law of Cosines to determine the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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5 Skill - Law of Cosines from figure · Level 2

Use the Law of Cosines to determine the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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6 Skill - Law of Cosines from figure · Level 2

Use the Law of Cosines to determine the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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7 Skill - Law of Cosines from figure · Level 2

Use the Law of Cosines to determine the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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8 Skill - Law of Cosines from figure · Level 2

Use the Law of Cosines to determine the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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9 Skill - Solve triangle ABC (figure) · Level 3

Solve triangle ABC as shown in the figure.

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10 Skill - Solve triangle ABC (figure) · Level 3

Solve triangle ABC as shown in the figure.

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11 Skill - Solve triangle ABC (SAS) · Level 2

Solve triangle ABC where \(a = 3.0\), \(b = 4.0\), \(\angle C = 53^{\circ}\).

12 Skill - Solve triangle ABC (SAS) · Level 2

Solve triangle ABC where \(b = 60\), \(c = 30\), \(\angle A = 70^{\circ}\).

13 Skill - Solve triangle ABC (SSS) · Level 2

Solve triangle ABC where \(a = 20\), \(b = 25\), \(c = 22\).

14 Skill - Solve triangle ABC (SSS) · Level 2

Solve triangle ABC where \(a = 10\), \(b = 12\), \(c = 16\).

15 Skill - Solve triangle ABC (SSA ambiguous) · Level 3

Solve triangle ABC where \(b = 125\), \(c = 162\), \(\angle B = 40^{\circ}\).

16 Skill - Solve triangle ABC · Level 3

Solve triangle ABC where \(a = 65\), \(c = 50\), \(\angle C = 52^{\circ}\).

17 Skill - Solve triangle ABC · Level 3

Solve triangle ABC where \(a = 50\), \(b = 65\), \(\angle A = 55^{\circ}\).

18 Skill - Solve triangle ABC (ASA) · Level 2

Solve triangle ABC where \(a = 73.5\), \(\angle B = 61^{\circ}\), \(\angle C = 83^{\circ}\).

19 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown. (Use either the Law of Sines or the Law of Cosines, as appropriate.)

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20 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown. (Use either the Law of Sines or the Law of Cosines, as appropriate.)

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21 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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22 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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23 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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24 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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25 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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26 Skill - Find indicated side or angle (figure) · Level 2

Find the indicated side \(x\) or angle \(\theta\) from the triangle shown.

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27 Skill - Heron's Formula · Level 2

Find the area of the triangle whose sides have lengths \(a = 9\), \(b = 12\), \(c = 15\).

28 Skill - Heron's Formula · Level 2

Find the area of the triangle whose sides have lengths \(a = 1\), \(b = 2\), \(c = 2\).

29 Skill - Heron's Formula · Level 2

Find the area of the triangle whose sides have lengths \(a = 7\), \(b = 8\), \(c = 9\).

30 Skill - Heron's Formula · Level 2

Find the area of the triangle whose sides have lengths \(a = 11\), \(b = 100\), \(c = 101\).

31 Skill - Area of shaded figure · Level 3

Find the area of the shaded figure shown, rounded to two decimals.

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32 Skill - Area of shaded figure · Level 3

Find the area of the shaded figure shown, rounded to two decimals.

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33 Skill - Area of shaded figure · Level 3

Find the area of the shaded figure shown, rounded to two decimals.

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34 Skill - Area of shaded figure · Level 3

Find the area of the shaded figure shown, rounded to two decimals.

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35 Application - Tangent circles · Level 4

Three circles of radii 4, 5, and 6 cm are mutually tangent. Find the shaded area enclosed between the circles.

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36 Skill - Proof (Projection Laws) · Level 4

Prove that in triangle ABC: \(a = b \cos C + c \cos B\) \(b = c \cos A + a \cos C\) \(c = a \cos B + b \cos A\) These are called the Projection Laws. [Hint: To get the first equation, add the second and third equations in the Law of Cosines and solve for \(a\).]

37 Application - Surveying · Level 3

To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information.

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38 Application - Geometry (parallelogram) · Level 3

A parallelogram has sides of lengths 3 and 5, and one angle is 50°. Find the lengths of the diagonals.

39 Application - Calculating distance · Level 3

Two straight roads diverge at an angle of 65°. Two cars leave the intersection at 2:00 P.M., one traveling at 50 mi/h and the other at 30 mi/h. How far apart are the cars at 2:30 P.M.?

40 Application - Calculating distance · Level 3

A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 40 mi/h, how far is it from its starting position?

41 Application - Dead reckoning · Level 3

A pilot flies in a straight path for 1 h 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 h in the new direction. If she maintains a constant speed of 625 mi/h, how far is she from her starting position?

42 Application - Navigation (two boats) · Level 3

Two boats leave the same port at the same time. One travels at a speed of 30 mi/h in the direction N 50° E and the other travels at a speed of 26 mi/h in a direction S 70° E (see the figure). How far apart are the two boats after one hour?

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43 Application - Navigation (bearings) · Level 3

A fisherman leaves his home port and heads in the direction N 70° W. He travels 30 mi and reaches Egg Island. The next day he sails N 10° E for 50 mi, reaching Forrest Island.

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(a) Find the distance between the fisherman's home port and Forrest Island.

Answer for 43(a)

(b) Find the bearing from Forrest Island back to his home port.

Answer for 43(b)

Enter your answer directly below each part above.

44 Application - Navigation (course correction) · Level 4

Airport B is 300 mi from airport A at a bearing N 50° E (see the figure). A pilot wishing to fly from A to B mistakenly flies due east at 200 mi/h for 30 minutes, when he notices his error.

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(a) How far is the pilot from his destination at the time he notices the error?

Answer for 44(a)

(b) What bearing should he head his plane in order to arrive at airport B?

Answer for 44(b)

Enter your answer directly below each part above.

45 Application - Triangular field · Level 2

A triangular field has sides of lengths 22, 36, and 44 yd. Find the largest angle.

46 Application - Towing a barge · Level 3

Two tugboats that are 120 ft apart pull a barge, as shown. If the length of one cable is 212 ft and the length of the other is 230 ft, find the angle formed by the two cables.

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47 Application - Flying kites · Level 2

A boy is flying two kites at the same time. He has 380 ft of line out to one kite and 420 ft to the other. He estimates the angle between the two lines to be 30°. Approximate the distance between the kites.

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48 Application - Securing a tower · Level 3

A 125-ft tower is located on the side of a mountain that is inclined 32° to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed.

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49 Application - Cable car · Level 4

A steep mountain is inclined 74° to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 800 ft from the base to the top of the mountain, as shown. Find the shortest length of cable needed.

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50 Application - CN Tower (partial / OCR truncated) · Level 3

The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is 43°. She also observes that the angle between the vertical and the line of sight to one of the landmarks is [problem text truncated in OCR].

51 Application - Land Value · Level 2

Land Value. Land in downtown Columbia is valued at \$20 a square foot. What is the value of a triangular lot with sides of lengths 112, 148, and 190 ft?

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52 Discovery/Discussion/Writing - Solving for Angles in a Triangle · Level 3

Solving for the Angles in a Triangle. The paragraph that follows the solution of Example 3 on page 477 explains an alternative method for finding \(\angle B\) and \(\angle C\), using the Law of Sines. Use this method to solve the triangle in the example, finding \(\angle B\) first and then \(\angle C\). Explain how you chose the appropriate value for the measure of \(\angle B\). Which method do you prefer for solving an SAS triangle problem, the one explained in Example 3 or the one you used in this exercise?

53 Example - Length of a Tunnel · Level 3

A tunnel is to be built through a mountain. To estimate the length of the tunnel, a surveyor makes the measurements shown in Figure 3: \(a = 388\) ft, \(b = 212\) ft, and \(\angle C = 82.4^{\circ}\). Use the surveyor's data to approximate the length of the tunnel.

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54 Example - SSS, the Law of Cosines · Level 3

The sides of a triangle are \(a = 5\), \(b = 8\), and \(c = 12\). Find the angles of the triangle.

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55 Example - SAS, the Law of Cosines · Level 3

Solve triangle \(A B C\), where \(\angle A = 46.5^{\circ}\), \(b = 10.5\), and \(c = 18.0\).

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56 Example - Navigation (bearing) · Level 3

A pilot sets out from an airport and heads in the direction N 20° E, flying at 200 mi/h. After one hour, he makes a course correction and heads in the direction N 40° E. Half an hour after that, engine trouble forces him to make an emergency landing.

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(a) Find the distance between the airport and his final landing point.

Answer for 56(a)

(b) Find the bearing from the airport to his final landing point.

Answer for 56(b)

Enter your answer directly below each part above.

57 Example - Heron's Formula · Level 2

A businessman wishes to buy a triangular lot in a busy downtown location. The lot frontages on the three adjacent streets are 125, 280, and 315 ft. Find the area of the lot.

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