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In this lesson, we'll learn how to solve problems involving three sides and one angle in a triangle. The Law of Cosines, a modification of the Pythagorean Theorem, will save the day.

Triangles

Triangles - they can be troublesome. Love triangles? That's a phrase you never really use to describe a good situation. And the Bermuda triangle? The risk of disappearing into another dimension kind of spoils the otherwise amazing Bermuda vacation.

Fortunately, we have all kinds of great tools to help us with triangles in math problems. Let's say we know two angles and one side length of a triangle. How do we find the missing side? The Law of Sines! That's a/sin A = b/sin B = c/sin C.

The Law of Sines is great for problems involving two angles and two sides. But what about problems involving three sides and one angle? There's another law for that!

Law of Cosines

It's called the Law of Cosines. Before you think, 'Hey, what's with all these laws? Isn't that kind of restricting? I need to be free, man!' Hold on. This is one of those helpful laws that makes your life easier. It's like a law that requires chocolate chip cookies to be tasty. It's in your best interest, trust me.

Anyway, the law of cosines involves a triangle like this, where our sides are labeled a, b and c. And then we have one angle, like A here.

The law states that a2 = b2 + c2 - 2bc(cosA). So the square of this side equals the sum of the squares of the other two sides minus the cosine of the angle opposite times twice the product of the two other sides.

We can modify the formula to fit whatever angle we have. So it could also be b2 = a2 + c2 - 2ac(cosB) or c2 = a2 + b2 - 2ab(cosC). Just match the corresponding angle and side so they're on opposite sides of the equation.

Does that look a little familiar? Remember the Pythagorean Theorem? a2 + b2 = c2. Notice that the Law of Cosines is the same basic thing, just adding that -2ab(cosC). Why? Because the Pythagorean Theorem only works on right triangles. So this modification accounts for the obliqueness of the triangle. An oblique triangle is just any triangle that isn't a right triangle.

But how can we possibly remember this? Well, the Pythagorean Theorem involves a, b and c, right? The Law of Cosines is the same thing, but with a second a, b and c. So our 2ab(cosC), is our second a, b and c.

Practice Problem #1

Let's practice. Here's a triangle where we know angle A is 112 degrees. If we know side b is 4 and side c is 15, what is the approximate length of side a?

What's our Law of Cosines? a2 = b2 + c2 - 2bc(cosA). Let's just plug in what we know: a2 = 42 + 152 - 2(4)(15)(cos112). 42 is 16. 152 is 225. Add those to get 241.

The cosine of 112 is about -.37. 2 * 4 is 8. 8 * 15 is 120. 120 * -.37 is -44.4. But since we're supposed to subtract -44.4 from 241, we just add it. That gets us 285.4. So a2 = 285.4. Don't forget that it's a2, not a. That means we need to take the square root of 285.4, which is about 17.

Quick double-check: if side b is 4 and side c is 15, does it make sense that side a is 17? Well, a is clearly the widest angle, so a should be the longest side. Yeah, this seems to make sense. So we're good!

Practice Problem #2

Let's try another. In this triangle, we know that side a is 12 and side c is 5. We also know that angle B is 33. What is the approximate length of side b?

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Remember, our Law of Cosines can be rearranged to suit our needs. We want to know side b, so we put that first. b2 = a2 + c2 - 2ac(cosB).

Now plug in what we know: b2 = 122 + 52 - 2(12)(5)(cos33). 122 is 144. 52 is 25. Add those to get 169. The cosine of 33 is about .84. 2 times 12 is 24. 24 times 5 is 120. 120 times .84 is 100.8. Let's now subtract 100.8 from 169. It's 68.2.

So b2 = 68.2. The square root of 68.2? It's about 8.

Let's double-check this one: if c is 5 and a is 12, does it look like b could be 8? Yeah, totally. Let's call this one done.

Practice Problem #3

How about one more? In this one, we know that side a is 14, side b is 12 and side c is 8. What is the measure of angle C?

This one is a little different. This time, we want to know an angle using the three sides. We can do this! Let's use our Law of Cosines: c2 = a2 + b2 - 2ab(cosC). If we plug in what we know, we get 82 = 142 + 122 - 2(14)(12)(cosC). Okay, 82 is 64. 142 is 196. 122 is 144. So, 196 + 144 is 340.

Let's subtract that from 64 to get -276. Now, 2 times 14 is 28. And 28 * 12 is 336.

So we have -276 = -336(cosC). Let's divide by -336 to get cosC = about .82. So the inverse cosine of .82 is about 35 degrees.

We could use the Law of Cosines to get the other two angles and be certain they add up to about 180. A far less scientific way to check our math is to just look at the triangle. Does angle C look like it's about 35 degrees? Yes!

Lesson Summary

In summary, the Law of Cosines is helpful with problems involving three sides and one angle in oblique triangles, or any triangle that doesn't contain a right angle.

The Law of Cosines is a2 = b2 + c2 - 2bc(cosA). However, we can rearrange the law to fit our needs. Always remember that the triangle side on the left of the equation should match the cosine angle on the right side.

The Law of Cosines is a variation on the Pythagorean Theorem, wherein we add a second a, b and c.

Learning Outcome

Once this lesson has been studied, you could be capable of writing the Law of Cosines and distinguishing between this law and the Pythagorean Theorem. You might have the skills necessary to use the Law of Cosines to solve triangles with three sides and one angle as you complete related practice equations.

The law of cosines is a technique applied to a triangle to find the rest of the sides and angles if two sides and the angle between them are given, or all three sides are given.

The two cases are:

SAS (side-angle-side )

and SSS (side-side-side)

Application

To see how we can practically use the law of cosines, we will find how a flight needs to revise the direction in order to correct a 10-degree error and get to the destination at the same time as scheduled.

In attempting to fly from Houston to San Antonio, a distance of 200 miles, a pilot took a course that was 10 degrees in error. If the aircraft maintains an average speed of 200 mph and if the error is discovered after 15 minutes, we will find out the angle the pilot should turn to head toward San Antonio.

For this, you will do the following.

1) Find the distance traveled by the plane until the error was discovered and draw a triangle to illustrate the path of the plane.

2) Find the third length of the triangle found in 1).

3) Find the angle of correction.

4) Find the new average speed that the pilot should maintain so that the total time of the trip is 60 minutes.

Solutions

1) The distance traveled by the plane flying at 200 mph for 15 minutes is 5 miles; the triangle with two sides, 200 and 5, and the angle between the two sides of 10 degrees is shown below

2) Applying the law of cosines, we obtain:

3) To find the angle of correction, we can use the law of sines or, again, the law of cosines to obtain

4) To travel the total distance in 60 minutes, the pilot needs to travel the BC distance in 45 minutes, so the speed is BC * 60/45 = 260.1 mph.

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