If we knew either or , we could use trigonometric ratios to find the height of the support beam. However, neither of these angle measures are given to us. Since we know all three sides of , we can use the Law of Cosines to find one of these angles. We will find .
Now that we know , we can use it to find the length of .
Answer: Yes, the builder’s estimate of for the support beam is accurate.
Points to Consider
How is the Pythagorean Theorem a special case of the Law of Cosines?
In the SAS case, is it possible to use the Law of Cosines to find all missing sides and angles?
In which cases can we not use the Law of Cosines? Explain.
Give an example of three side lengths that do not form a triangle.
Lesson Summary
The Law of Cosines is used in oblique (non-right triangles) because we cannot use the Theorem of Pythagoras or trigonometric ratios.
We can use the Law of Cosines when we know two sides and the included angle in a triangle (SAS). This allows us to find the third side of the triangle.
We can also use the Law of Cosines when we know all three sides in a triangle (SSS). This enables us to find any or all three of the angles in the triangle.
The Law of Cosines can be used to verify the oblique triangles are accurately drawn.
There are many real-world situations in which the Law of Cosines is used. The Law of Cosines comes in handy when measurements are hard to obtain or are not reliable due to uneven surfaces.
Review Questions
Using each figure and the given information below, decide which side(s) or angle(s) you could find using the Law of Cosines.
Given Information Figure What can you find?
a.
b.
c.
d.
e.
f.
Using the figures and information from the chart above, use the Law of Cosines to find the following: side
the largest angle
side
the smallest angle
side
the second largest angle
In , , and . Find the measure of all three angles.
Find using the Theorem of Pythagoras, Law of Cosines, trig functions, or any combination of the three.
Find using the Theorem of Pythagoras, Law of Cosines, trig functions, or any combination of the three if , and .
Use the Law of Cosines to determine whether or not the following triangles are drawn accurately. Is drawn accurately? If not, determine how much is off by.
Is drawn accurately? If not, determine how much side is off by.
A businessman is traveling down Interstate and has intermittent cell phone service. There is a transmission tower near Interstate . The range of service from the tower forms a angle and the range of service is to one section of and to another point on . If the businessman is traveling at a speed of per hour, how long will he have service for?
If he slows down to , how much longer will he be able to have service?
A dock is being built so that it is away from one buoy and away from a second buoy. The two buoys are apart. What angle does the dock form with the two buoys?
If the second buoy is moved so that it is away from the dock and away from the first buoy, how does this affect the angle formed by the dock and the two buoys?
An artist is making a large sculpture for the lobby of a new building. She has drawn out what she wants the sculpture to look like at the left. If she wants , , and , verify that the length of would be . If not, figure out the correct measure.
A golfer hits the ball from the tee. His shot is a hook (curves to the left) from the path straight to the flag on the green. (a) If the tee is from the flag, how far is the ball away from the flag? (b) If the golfer’s next shot is and is hooked from the path straight to the flag, how far is ball away now?
Given the numbers , and write a problem that uses the Law of Cosines.
The sides of a triangle are , and . What is its area?
Hint: Use the Law of Cosines then use some right triangle trig.
A person inherits a triangular piece of land with dimensions , , and . What is the area of the piece of land? How much of an acre is it?
A family’s farm plot is a quadrilateral with dimensions: the longest side is and the shortest side is . The side opposite the long side is . The shorter diagonal is . What is the area of the land in square feet?
The height of a crane off the ground is determined by the steel cable from the drum it wraps around. The lowest angle with the vertical the crane can make is . The largest angle the crane can make with the ground is . The length of the crane boom is . The distance from the base of the crane to the edge of the drum is . How long is the cable at the crane’s lowest reach?
How long is the cable at the crane’s highest reach?
A biomechanics class is designing a functioning artificial arm for an adult. They are using a hydraulic cylinder (fluid filled) to be the bicep’s muscle. The elbow is at point . The forearm dimension is . The upper arm dimension is . The cylinder attaches from the top of the upper arm at point and to a point on the lower arm from the mechanical elbow at point . When fluid is pumped out of the cylinder the distance is shortened. The forearm goes up raising the hand at point .
Some fluid is pumped out of the cylinder to make the distance shorter. What is the new angle of the arm,
Review Answers
side
and
side
and
side
.
.
is off by
side is off by
He will have service for .
He will have service for longer.
The angle formed is .
The angle will need to be rather than or less.
The length of would need to be , not .
The ball is away from the flag.
His second shot is away from the flag.
Student answers will vary.
The area is
The area of the land is or .
The area of the land is .
The cable is at the crane’s lowest reach and at the crane’s highest reach.
The new angle of the arm is .
Supplemental Links
http://math.boisestate.edu/~tconklin/MATH144/Main/Extras/Law%20of%20Cosines%202.pps
Vocabulary
Law of Cosines
A general statement relating the lengths of the sides of a general triangle to the cosine of one of its angles.
oblique triangle
A non-right triangle.
Area of a Triangle
Learning Objectives
A student will be able to:
Understand how the area formula is derived.
Apply the area formula to triangles where you know two sides and the included angle (SAS).
Apply the are formula to triangles where you know all three sides (SSS).
Understand Heron’s Formula.
Use the area formulas in real-world and applied problems.
Introduction
Real-World Application: The Pyramid Hotel recently installed a triangular pool. One side of the pool is , another side is , and the angle in between the two sides is . If the hotel manager needs to order a cover for the pool, and the cost is per square foot, how much can he expect to spend?
In this situation, we need to find the area of the surface of the pool in order to calculate the cost of the cover. We have already learned that the formula for the area of a triangle is where is the base of the triangle and is the height. The problem with this formula is that it can only be used when the height of the triangle is known. In this situation, we don’t know the height of the triangle formed by the sides of the pool. How do we find the area if we don’t know the height?
We will refer back to this application later on.
In this section, we will look at how we can derive a new formula using the area formula that we already kn
ow and the sine function. This new formula will allow us to find the area of a triangle when we don’t know the height. We will also look at when we can use this formula and how to apply it to real-world situations.
Derive Area = 1/2 bcsinA
We can use the area formula from above , as well as the sine function, to derive a new formula that can be used when the height is unknown.
In at the right, is altitude from to . We will refer to the length of as since It represents the height of the triangle. Also, we will refer to the area of the triangle as to avoid confusing the area with
We can use a similar method to derive all three forms of the area formula:
Find the Area Using Two Sides and an Included Angle--SAS (side-angle-side)
The formula requires us to know two sides and the included angle (SAS) in a triangle. Once we know these three things, we can easily calculate the area of an oblique triangle.
Example 1:
In , , , and . Find the area of the triangle.
Answer: The area of the triangle is approximately
We will now refer back to the application at the beginning of the chapter.
In order to find the cost of the cover, we first need to know the area of the cover. Once we know how many square feet the cover is, we can calculate the cost.
In the illustration above, you can see that we know two of the sides and the included angle. This means we can use the formula .
Answer: The area of the pool cover is The cost of the cover will be
Find the Area Using Three Sides--SSS (side-side-side) Heron’s Formula
In the last section, we learned how to find the area of an oblique triangle when we know two sides and the included angle using the formula . We could also find the area of a triangle in which we know all three sides by first using the Law of Cosines to find one of the angles and then using the formula . While this process works, it is time-consuming and requires a lot of calculation. Fortunately, we have another formula, called Heron’s Formula, which allows us to calculate the area of a triangle when we know all three sides.
Heron’s Formula:
where or half of the perimeter of the triangle.
Example 2:
In , and . Find the area of the triangle.
Answer: The area is approximately .
Real-World Application:
Tile: A handyman is installing a tile floor in a kitchen. Since the corners of the kitchen are not exactly square, he needs to have special triangular shaped triangles made for the corners. One side of the tile needs to be , the second side needs to be ,and the third side is If the tile costs per square foot, and he needs four of them, how much will it cost to have the tiles made?
In order to find the cost of the tiles, we first need to find the area of one tile. Since we know the measurements of all three sides, we can use Heron’s Formula to calculate the area.
The area of one tile would be The cost of the tile is given to us in square feet, while the area of the tile is in square inches. In order to find the cost of one tile, we must first convert the area of the tile into square feet.
Answer: The cost for four tiles would be .
Applications, Technological Tools
We have already looked at two examples of situations where we can apply the two new area formulas we learned in this section. In this section, we will look at another real-world application where we know the area but need to find another part of the triangle, as well as an application involving a quadrilateral.
Real-World Application:
The jib sail on a sailboat came untied and the rope securing it was lost. If the area of the jib sail is , use the figure and information at the right to find the length of the rope.
Since we know the area, one of the sides, and one angle of the jib sale, we can use the formula to find the side of the jib sale that is attached to the mast. We will call this side .
Now that we know side , we know two sides and the included angle in the triangle formed by the mast, the rope, and the jib sail. We can now use the Law of Cosines to calculate the length of the rope.
Answer: The length of the rope is approximately .
Quadrilaterals: In quadrilateral at the right, The area of , the area of , and . Find the perimeter of
In order to find the perimeter of , we need to know sides and . Since we know the area, one side, and one angle in each of the triangles, we can use to figure out and .
Now that we know and , we know two sides and the included angle in each triangle (SAS). This means that we can use the Law of Cosines to find the other two sides, and . First we will find .
Now, we will find .
Finally, we can calculate the perimeter since we have found all four sides of the quadrilateral.
Answer: The perimeter of is .
Points to Consider
Why can’t s (half of the perimeter) in Heron’s Formula be smaller than any of the three sides in the triangle?
How could we find the area of a triangle is AAS, SSA, and ASA cases?
Is it possible to figure out the length of the third side of a triangle if we know the other two sides and the area?
Lesson Summary
In an oblique triangle where we know two sides and the included angle, we can use the formula to calculate the area of the triangle.
In an oblique triangle where we know all three sides of the triangle, we can calculate the area using Heron’s Formula.
Given the area, we can use these two area formulas to find an unknown side or angle.
We have explored three scenarios where we can use these area formulas in real-world situations. We will look at more applications in the review questions.
Review Questions
Using the figures and given information below, determine which formula you would need to use in order to find the area of each triangle (, , or Heron’s Formula).
Given Figure Formula
a. and
b. and
c.
d.
Find the area of all of the triangles in the chart above to the nearest tenth.
Using the given information and the figures below, decide which area formula you would need to use to find each side, angle, or area.
Given Figure Find Formula
a.
b.
c.
Using the figures and information from the table above, find the angle, side, or area requested.
The Pyramid Hotel is planning on repainting the exterior of the building. The building has four sides that are isosceles triangles with bases measuring and legs measuring . What is the total area that needs to be painted?
If one gallon of paint covers , how many gallons of paint are needed?
A contractor needs to replace a triangular section of roof on the front of a house. The sides of the triangle are , and . If one bundle of shingles covers and costs , how many bundles does he need to purchase? How much will the shingles cost him? How much of the bundle will go to waste?
A farmer needs to replant a triangular section of crops that died unexpectedly. One side of the triangle measures , another measures , and the angle formed by these two sides is . What is the area of the section of crops that needs to be replanted?
The farmer goes out a few days later to discover that more crops have died. The side that used to measure now measures . How much has the area that needs to be replanted increased by?
Find the perimeter of the quadrilateral at the left If the area of and the area of .
In is an altitude from to . The area of , and . Find .
Show that in any triangle , .
Review Answers
Heron’s formula
The total area is
of paint are needed.
He will need . The shingles will cost him . will go to waste.
The area that needs to be replaced is
The area has increased by
The perimeter of the quadrilateral is .
is approximately .
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Supplemental Links
http://www.mste.uiuc.edu/dildine/heron/triarea.html
Vocabulary
Heron’s Formula
A formula used to calculate the area of a triangle when all three sides are known.
The Law of Sines
Learning Objectives
A student will be able to:
Understand how both forms of the Law of Sines are obtained.
Apply the Law of Sines when you know two angles and a non-included side (AAS).
Apply the Law of Sines if you know two angles and the included side (ASA).
CK-12 Trigonometry Page 20
