Ships and Planes Precalculus Key Drawing

Learning Objectives

In this department, you will:

• Use the Law of Cosines to solve oblique triangles.
• Solve practical problems using the Law of Cosines.
• Use Heron'south formula to find the surface area of a triangle.

Suppose a gunkhole leaves port, travels 10 miles, turns xx degrees, and travels another 8 miles equally shown in Effigy 1. How far from port is the boat?

Figure 1

Unfortunately, while the Law of Sines enables us to accost many not-right triangle cases, it does not help the states with triangles where the known angle is betwixt two known sides, a SAS (side-bending-side) triangle, or when all three sides are known, but no angles are known, a SSS (side-side-side) triangle. In this department, nosotros volition investigate some other tool for solving oblique triangles described by these last two cases.

Using the Constabulary of Cosines to Solve Oblique Triangles

The tool we need to solve the problem of the boat's altitude from the port is the Police force of Cosines, which defines the human relationship among angle measurements and side lengths in oblique triangles. Three formulas make up the Law of Cosines. At first glance, the formulas may appear complicated because they include many variables. However, once the pattern is understood, the Police force of Cosines is easier to work with than most formulas at this mathematical level.

Agreement how the Police of Cosines is derived will be helpful in using the formulas. The derivation begins with the Generalized Pythagorean Theorem, which is an extension of the Pythagorean Theorem to not-correct triangles. Here is how it works: An arbitrary non-right triangle $ABC$ is placed in the coordinate aeroplane with vertex $A$ at the origin, side $c$ drawn along the 10-axis, and vertex $C$ located at some bespeak $\left(x,y\right)$ in the aeroplane, as illustrated in Figure 2. Generally, triangles exist anywhere in the aeroplane, simply for this caption we will place the triangle every bit noted.

Figure two

We can drop a perpendicular from $C$ to the x-centrality (this is the altitude or height). Recalling the bones trigonometric identities, nosotros know that

$$\cos \theta =\frac{x\text{(adjacent)}}{b\text{(hypotenuse)}}\text{ and}\sin \theta =\frac{y\text{(opposite)}}{b\text{(hypotenuse)}}$$

In terms of $\theta ,x=b\cos \theta$ and $y=b\sin \theta .$ The $(x,y)$ signal located at $C$ has coordinates $(b\cos \theta ,$ $b\sin \theta ).$ Using the side $\left(x-c\right)$ as one leg of a correct triangle and $y$ as the 2nd leg, nosotros tin find the length of hypotenuse $a$ using the Pythagorean Theorem. Thus,

$$\begin{array}{llllll}{a}^2={(\mathrm{ten}-c)}^2+y^2\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ ={(b\cos \theta -c)}^2+{(b\sin \theta )}^2\hfill & \hfill & \hfill & \hfill & \hfill & \text{Substitute}(b\cos \theta )\text{ for}\mathrm{10}\text{and}(b\sin \theta )\text{for}y.\hfill \\ =\left(b^2{\cos }^2\theta -\mathrm{two}bc\cos \theta +c^2\right)+b^{\mathrm{two}}{\sin }^2\theta \hfill & \hfill & \hfill & \hfill & \hfill & \text{Expand the perfect square}.\hfill \\ =b^{\mathrm{ii}}{\cos }^{\mathrm{ii}}\theta +b^{\mathrm{ii}}{\sin }^{\mathrm{ii}}\theta +c^2-2bc\cos \theta \hfill & \hfill & \hfill & \hfill & \hfill & \text{Grouping terms noting that}{\cos }^2\theta +{\sin }^2\theta =1.\hfill \\ =b^2\left({\cos }^2\theta +{\sin }^{\mathrm{two}}\theta \right)+c^2-\mathrm{two}bc\cos \theta \hfill & \hfill & \hfill & \hfill & \hfill & \text{Factor out}{b}^2.\hfill \\ a^2=b^{\mathrm{ii}}+c^{\mathrm{ii}}-2bc\cos \theta \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$$

The formula derived is 1 of the iii equations of the Law of Cosines. The other equations are plant in a similar way.

Keep in mind that it is always helpful to sketch the triangle when solving for angles or sides. In a real-world scenario, attempt to draw a diagram of the situation. Every bit more information emerges, the diagram may have to be altered. Make those alterations to the diagram and, in the end, the problem will be easier to solve.

Police of Cosines

The Constabulary of Cosines states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other ii sides and the cosine of the included bending. For triangles labeled as in Figure 3, with angles $\alpha ,\beta ,$ and $\gamma ,$ and contrary respective sides $a,b,$ and $c,$ respectively, the Police force of Cosines is given as iii equations.

$$\begin{array}{l}{a}^2=b^{\mathrm{ii}}+c^{\mathrm{two}}-2bc\cos \alpha \\ b^2=a^{\mathrm{ii}}+c^2-\mathrm{ii}ac\cos \beta \\ c^2=a^{\mathrm{ii}}+b^2-\mathrm{ii}ab\cos \gamma \end{array}$$

Figure iii

To solve for a missing side measurement, the corresponding opposite angle measure is needed.

When solving for an angle, the corresponding opposite side measure is needed. We can employ another version of the Police force of Cosines to solve for an angle.

$$\begin{array}{l}\hfill \\ \begin{array}{l}\begin{array}{l}\hfill \\ \cos \alpha =\frac{b^2+c^2-a^{\mathrm{ii}}}{2bc}\hfill \end{array}\hfill \\ \cos \beta =\frac{a^{\mathrm{two}}+c^{\mathrm{ii}}-b^2}{2ac}\hfill \\ \cos \gamma =\frac{a^2+b^2-c^2}{\mathrm{two}ab}\hfill \end{array}\hfill \end{array}$$

How To

Given two sides and the angle betwixt them (SAS), find the measures of the remaining side and angles of a triangle.

1. Sketch the triangle. Identify the measures of the known sides and angles. Employ variables to correspond the measures of the unknown sides and angles.
2. Apply the Law of Cosines to find the length of the unknown side or angle.
3. Apply the Law of Sines or Cosines to notice the measure of a second bending.
4. Compute the measure of the remaining angle.

Example i

Finding the Unknown Side and Angles of a SAS Triangle

Find the unknown side and angles of the triangle in Figure 4.

Figure four

Try Information technology #ane

Find the missing side and angles of the given triangle: $\alpha =\mathrm{30°},$ $b=12,$ $c=24.$

Example two

Solving for an Angle of a SSS Triangle

Detect the bending $\alpha$ for the given triangle if side $a=\mathrm{twenty},$ side $b=25,$ and side $c=18.$

Assay

Because the inverse cosine can return any angle between 0 and 180 degrees, in that location will not be any cryptic cases using this method.

Endeavour It #2

Given $a=\mathrm{v},b=7,$ and $c=10,$ find the missing angles.

Solving Applied Problems Using the Law of Cosines

Merely as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is applicative to situations in which the given information fits the cosine models. We may see these in the fields of navigation, surveying, astronomy, and geometry, simply to name a few.

Case 3

Using the Law of Cosines to Solve a Communication Problem

On many cell phones with GPS, an gauge location tin be given before the GPS signal is received. This is achieved through a procedure called triangulation, which works by using the distances from two known points. Suppose at that place are two cell telephone towers within range of a cell telephone. The two towers are located 6000 feet apart along a directly highway, running east to due west, and the cell phone is northward of the highway. Based on the signal filibuster, it can be adamant that the bespeak is 5,050 feet from the first tower and 2,420 anxiety from the second belfry. Decide the position of the cell phone north and east of the first belfry, and make up one's mind how far it is from the highway.

Case 4

Calculating Distance Traveled Using a SAS Triangle

Returning to our trouble at the beginning of this section, suppose a boat leaves port, travels ten miles, turns 20 degrees, and travels some other viii miles. How far from port is the boat? The diagram is repeated hither in Figure 8.

Figure 8

Using Heron's Formula to Find the Surface area of a Triangle

We already learned how to find the area of an oblique triangle when we know two sides and an bending. We as well know the formula to observe the expanse of a triangle using the base and the height. When nosotros know the three sides, however, we tin can utilize Heron's formula instead of finding the height. Heron of Alexandria was a geometer who lived during the first century A.D. He discovered a formula for finding the area of oblique triangles when three sides are known.

Heron'due south Formula

Heron'southward formula finds the expanse of oblique triangles in which sides $a,b\text{,}$ and $c$ are known.

$$\text{Area}=\sqrt{s\left(s-a\right)\left(\mathrm{due\; south}-b\right)\left(\mathrm{due\; south}-c\right)}$$

where $s=\frac{\left(a+b+c\right)}{2}$ is one half of the perimeter of the triangle, sometimes chosen the semi-perimeter.

Case five

Using Heron'due south Formula to Detect the Area of a Given Triangle

Find the surface area of the triangle in Figure 9 using Heron's formula.

Effigy 9

Effort Information technology #iii

Use Heron's formula to notice the area of a triangle with sides of lengths $a=29.7\text{ft},b=42.3\text{ft},$ and $c=38.4\text{ft}.$

Example half dozen

Applying Heron's Formula to a Real-World Trouble

A Chicago city programmer wants to construct a edifice consisting of artist'southward lofts on a triangular lot bordered by Blitz Street, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, forth Wabash Artery it is approximately 43.5 meters, and along Pearson Street information technology is approximately 34.1 meters. How many square meters are available to the programmer? Run across Figure 10 for a view of the city property.

Figure 10

Endeavor It #4

Find the area of a triangle given $a=\mathrm{iv.38}\text{ft},b=3.79\text{ft,}$ and $c=5.22\text{ft}\text{.}$

eight.2 Section Exercises

Verbal

ane.

If yous are looking for a missing side of a triangle, what do you need to know when using the Police of Cosines?

two .

If yous are looking for a missing bending of a triangle, what practice you need to know when using the Police of Cosines?

three.

Explain what $s$ represents in Heron'due south formula.

four .

Explain the relationship between the Pythagorean Theorem and the Police force of Cosines.

5.

When must yous use the Law of Cosines instead of the Pythagorean Theorem?

Algebraic

For the following exercises, assume $\alpha$ is contrary side $a,\beta$ is opposite side $b,$ and $\gamma$ is contrary side $c.$ If possible, solve each triangle for the unknown side. Round to the nearest tenth.

six .

$\gamma =\mathrm{41.2°},a=\mathrm{ii.49},b=\mathrm{3.xiii}$

seven.

$\alpha =\mathrm{120°},b=6,c=7$

8 .

$\beta =\mathrm{58.7°},a=10.6,c=\mathrm{xv.7}$

9.

$\gamma =\mathrm{115°},a=18,b=23$

10 .

$\alpha =\mathrm{119°},a=26,b=14$

11.

$\gamma =\mathrm{113°},b=10,c=32$

12 .

$\beta =\mathrm{67°},a=49,b=38$

13.

$\alpha =\mathrm{43.1°},a=\mathrm{184.two},b=\mathrm{242.viii}$

14 .

$\alpha =\mathrm{36.half-dozen°},a=\mathrm{186.ii},b=242.2$

15.

$\beta =\mathrm{50°},a=105,b=45{}_{}{}^{}$

For the post-obit exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

16 .

$a=42,b=19,c=\mathrm{thirty};$ find angle $A.$

17.

$a=\mathrm{fourteen},b=13,c=20;$ notice angle $C.$

18 .

$a=16,b=31,c=\mathrm{xx};$ find angle $B.$

19.

$a=13,b=22,c=28;$ detect angle $A.$

20 .

$a=108,b=132,c=160;$ observe angle $C.$

For the following exercises, solve the triangle. Round to the nearest tenth.

21.

$A=\mathrm{35°},b=8,c=11$

22 .

$B=\mathrm{88°},a=\mathrm{four.4},c=5.2$

23.

$C=\mathrm{121°},a=21,b=37$

24 .

$a=13,b=11,c=\mathrm{xv}$

25.

$a=\mathrm{3.i},b=\mathrm{three.v},c=\mathrm{v}$

26 .

$a=51,b=25,c=29$

For the following exercises, use Heron's formula to detect the area of the triangle. Round to the nearest hundredth.

27.

Find the area of a triangle with sides of length xviii in, 21 in, and 32 in. Round to the nearest tenth.

28 .

Find the area of a triangle with sides of length 20 cm, 26 cm, and 37 cm. Round to the nearest tenth.

29.

$a=\frac{1}{2}\text{m},b=\frac{1}{3}\text{m},c=\frac{\mathrm{one}}{\mathrm{four}}\text{m}$

30 .

$a=12.4\text{ ft},b=13.7\text{ ft},c=\mathrm{20.ii}\text{ ft}$

31.

$a=1.6\text{ yd},b=2.6\text{ yd},c=4.1\text{ yd}$

Graphical

For the following exercises, find the length of side $x.$ Circular to the nearest tenth.

32 .

34 .

36 .

For the following exercises, observe the measurement of angle $A.$

38 .

40 .

42 .

Find the measure of each angle in the triangle shown in Figure 11. Circular to the nearest tenth.

Figure 11

For the post-obit exercises, solve for the unknown side. Round to the nearest tenth.

44 .

46 .

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

48 .

50 .

Extensions

52 .

A parallelogram has sides of length xvi units and 10 units. The shorter diagonal is 12 units. Find the measure of the longer diagonal.

53.

The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 anxiety. Discover the length of the shorter diagonal.

54 .

The sides of a parallelogram are 28 centimeters and 40 centimeters. The measure out of the larger angle is 100°. Find the length of the shorter diagonal.

55.

A regular octagon is inscribed in a circle with a radius of 8 inches. (Meet Figure 12.) Find the perimeter of the octagon.

Effigy 12

56 .

A regular pentagon is inscribed in a circle of radius 12 cm. (See Effigy xiii.) Find the perimeter of the pentagon. Round to the nearest 10th of a centimeter.

Effigy thirteen

For the post-obit exercises, suppose that ${\mathrm{10}}^2=25+36-60\cos \left(52\right)$ represents the relationship of three sides of a triangle and the cosine of an angle.

58 .

Observe the length of the third side.

For the following exercises, find the area of the triangle.

lx .

Real-World Applications

62 .

A surveyor has taken the measurements shown in Figure 14. Find the distance across the lake. Round answers to the nearest 10th.

Effigy 14

63.

A satellite calculates the distances and angle shown in Figure 15 (not to scale). Find the distance betwixt the two cities. Round answers to the nearest tenth.

Figure 15

64 .

An airplane flies 220 miles with a heading of 40°, and then flies 180 miles with a heading of 170°. How far is the plane from its starting point, and at what heading? Round answers to the nearest tenth.

65.

A 113-foot tower is located on a hill that is inclined 34° to the horizontal, as shown in Figure 16. A guy-wire is to be attached to the pinnacle of the tower and anchored at a betoken 98 feet uphill from the base of operations of the belfry. Observe the length of wire needed.

Figure 16

66 .

Ii ships left a port at the same time. One send traveled at a speed of 18 miles per hour at a heading of 320°. The other ship traveled at a speed of 22 miles per hour at a heading of 194°. Detect the altitude between the two ships after 10 hours of travel.

67.

The graph in Figure 17 represents 2 boats parting at the same time from the same dock. The first boat is traveling at 18 miles per hr at a heading of 327° and the 2nd gunkhole is traveling at 4 miles per hour at a heading of threescore°. Observe the distance betwixt the two boats later 2 hours.

Effigy 17

68 .

A triangular swimming pool measures 40 anxiety on 1 side and 65 feet on some other side. These sides form an angle that measures 50°. How long is the third side (to the nearest tenth)?

69.

A pilot flies in a straight path for 1 hour xxx min. She then makes a class correction, heading 10° to the correct of her original course, and flies ii hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

70 .

Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is two,451 miles from Los Angeles. Draw a triangle connecting these three cities, and find the angles in the triangle.

71.

Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.

72 .

Two planes go out the aforementioned airdrome at the aforementioned time. One flies at 20° due east of due north at 500 miles per hour. The 2d flies at 30° due east of south at 600 miles per hour. How far apart are the planes after 2 hours?

73.

Two airplanes take off in unlike directions. One travels 300 mph westward and the other travels 25° north of due west at 420 mph. After 90 minutes, how far apart are they, bold they are flight at the aforementioned altitude?

74 .

A parallelogram has sides of length 15.4 units and 9.8 units. Its area is 72.9 foursquare units. Detect the measure of the longer diagonal.

75.

The four sequential sides of a quadrilateral accept lengths 4.5 cm, 7.9 cm, 9.iv cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?

76 .

The iv sequential sides of a quadrilateral take lengths 5.7 cm, 7.2 cm, nine.4 cm, and 12.8 cm. The angle between the two smallest sides is 106°. What is the area of this quadrilateral?

77.

Find the area of a triangular piece of country that measures 30 anxiety on one side and 42 anxiety on another; the included bending measures 132°. Circular to the nearest whole square foot.

78 .

Find the area of a triangular piece of land that measures 110 feet on ane side and 250 feet on some other; the included angle measures 85°. Round to the nearest whole square foot.

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Source: https://openstax.org/books/precalculus/pages/8-2-non-right-triangles-law-of-cosines

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