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Similarity, Proof, and Trigonometry

Solving Triangles Using the Law of Cosines

In previous lessons, it has been seen how trigonometric ratios can be used to solve right triangles. It has also been seen how the Law of Sines can be used to solve non-right triangles. However, there are cases in which this law is not useful. This lesson will explore the relationship between the side lengths and an angle measure of different triangles.

Catch-Up and Review

Here is a bundle of recommended readings before getting started with this lesson.

Try your knowledge on these topics.

Consider the triangle with vertices A, B, and C.

triangle
a Find the measure of the missing angle.
b Find the length of the longest side. If necessary, round the answer to the nearest integer.
c If the altitude of the triangle is 6, find AD and DB. If necessary, round the answer to the nearest integer.
triangle

Theory

Investigate Distances in Space Using Trigonometry

On a June night, Zosia observes an astronomical asterism, which is called the Summer Triangle.

From an observer's point of view, the apparent distances of celestial objects in the sky are measured in angular distance. The apparent distance from Altair to Deneb is 38 angular units and 34 angular units from Altair to Vega. Furthermore, the angle measure between these sides is Help Zosia find the apparent distance between Deneb and Vega.

Explore

Limitations of the Law of Sines

The applet below shows two different cases for a triangle with vertices A, B, and C. Case I shows the lengths of two sides and the measure of their included angle. Case II shows the lengths of all three sides.

What does the Law of Sines state? Is it possible to find the missing side lengths and angle measures by using the Law of Sines? Explain.

Discussion

Law of Cosines

When the Law of Sines cannot be used to solve triangles, the Law of Cosines may be applied.

Consider ABC with sides of length a, b, and c, which are respectively opposite the angles with measures A, B, and C.

triangle with angles and sides labeled

The following equations hold true with regard to ABC.

Proof

For Acute Angles
The first equation will be proven. The other two equations can be proven by following the same procedure.
Begin by drawing the altitude from B to its opposite side AC.
acute triangle and its height
By the definition of an altitude, both ADB and CDB are right triangles. By applying the Pythagorean Theorem to CDB and ADB, two equations can be obtained.
The binomial in Equation I can be expanded.
a2=(bx)2+h2
a2=b22bx+x2+h2
Notice that Equation II says that x2+h2 is equal to c2. Therefore, the expanded form of Equation I can be rewritten by using the Substitution Property of Equality.
a2=b22bx+x2+h2
a2=b22bx+c2
a2=b2+c22bx
Now, the x-term in this equation can be written using the cosine ratio.
acute angle of the triangle marker
In ADB, the cosine of A is the ratio of x to c.
Finally, can be substituted for x into a2=b2+c22bx. By doing so, the formula for the Law of Cosines is obtained.

Proof

For Obtuse Angles
The first equation will be proven for obtuse angles. The remaining equations can be proven similarly.
Consider ABC with side lengths of a, b, and c, respectively opposite the angles with measures A, B, and C, such that mA is greater than
obtuse triangle

The altitude of the triangle is the perpendicular segment from B to the extension of the base AC. Let D be the endpoint of this segment and x be the distance from D to A.

obtuse triangle and its height
From the definition of an altitude, it follows that BDA and BDC are right triangles. Two equations can be obtained by applying the Pythagorean Theorem to both triangles.
Expand the binomial in Equation I.
a2=h2+(b+x)2
a2=h2+b2+2bx+x2
a2=h2+x2+b2+2bx
From Equation II, h2+x2 is equal to c2. Therefore, the expanded form of Equation I can be rewritten by using the Substitution Property of Equality.
a2=h2+x2+b2+2bx
a2=c2+b2+2bx
Note that A and are supplementary angles. Using the cosine ratio of then gives an expression for the x-term.
obtuse triangle and angle A prime marked
In BDA, the cosine of is the ratio of x to c.
By the Sine and Cosine of Supplementary Values Angles, and have opposite values.
Finally, by substituting for x into a2=b2+c2+2bx, the Law of Cosines is obtained.
a2=b2+c2+2bx

Example

Knowing Two Sides and the Included Angle of a Triangle

In a triangle, when the lengths of two sides and the measure of their included angle are known, the missing side length can be found by applying the Law of Cosines.

Kriz wants to determine the distance between two trees on the other side of the river. Kriz uses a tool that measures the distances to objects. The tool is able to find the distances to each tree as 4 meters and 3.5 meters.

The angle between these sides, from where the Kriz stands with the measuring tool, measures Find the distance between the trees, and round the answer to the nearest tenth of a meter.

Hint

Start by naming the vertices and sides of the triangle.

Solution

For simplicity, the vertices and sides of the triangle will be named.

The Law of Cosines relates the lengths of the sides and the cosine of one of the angles. Therefore, the missing side length of the triangle can be found by using this law.
The next step is to substitute b=4, c=3.5, and into the equation. Then, solve for a.
Solve for a
Note that only the principal root was considered because a side length cannot be negative. Therefore, the distance between the trees is about 4.2 meters.

Example

Knowing Three Sides of a Triangle

When all the three side lengths of a triangle are known, the Law of Cosines can be used to find the measure of the angles.

Ramsha lives near a lighthouse. As she likes to observe the landscape, she notices that the light rays coming out of the lighthouse create an angle. She decides to ask the lighthouse keeper, but he insists on not telling her the measure of the angle. She sees a blueprint on the desk behind him, and quickly writes the lengths shown in the diagram before the grumpy keeper blocks her view!
Help Ramsha calculate the measure of each angle of the triangle formed by the light rays. Round the answers to the nearest degree.

Hint

To find mL, use the Law of Cosines. Then, the measures of other two angles can be found by using either the Law of Cosines or the Law of Sines.

Solution

As the diagram indicates, the light rays form a triangle. In this triangle, the three side lengths are known.

Triangle KLM

The measure of the three angles will be found one at a time.

Finding mL

To find the measure of L, the Law of Cosines can be used.
Substitute 35 for 105 for m, and 130 for k into the equation.
Solve for
Now, take the inverse cosine of both sides of the equation.

Evaluate right-hand side
The angle L measures about
Triangle KLM

Finding mM

Since the ratio of the sine of an angle to the length of its opposite side is constant, the following proportion can be written.
The equation can be solved for M.
Solve for M

The measure of M is about
Triangle KLM

Finding mK

By the Triangle Angle Sum Theorem, the sum of the interior angles of a triangle is
Therefore, all measures of KLM were found.
Triangle KLM
She did it! Ramsha found the measure of the angle to be .

Explore

Investigating the Cosine Ratio of Obtuse Angles

In ABC, all three side lengths and the measure of the angle at C are given. Examine how the length of AB changes as the measure of C varies.

Calculate the sum of the squares of AC and BC. Compare it with the square of AB when ACB is an acute, right, and obtuse angle. What can be concluded about the cosine ratio of obtuse angles?

Discussion

Relationship Between the Angles of a Triangle

Consider ABC, in which all three sides lengths and the measure of the angle at C are known. Let a, b, and c be the lengths of the sides opposite A, B, and C, respectively. In the following applet, the values of a2+b2 and c2 are shown. Move the slider to change the measure of C.
The table below shows the equation for the Law of Cosines when ACB is an acute, right, and obtuse angle. In the table it is also shown the relationship between a2+b2 and c2 for these values, and conclusions are made.
mC Relationship between a2+b2 and c2 Conclusion
c2<a2+b2 If C is acute, not too many conclusions can be made. The opposite side to C can be the largest side, the shortest side, or none.
c2=a2+b2 The cosine of is 0. In this case, the Law of Cosines becomes the Pythagorean Theorem. This means that the opposite side to C is the largest side of the triangle.
c2>a2+b2 If C is obtuse, its measure is greater than the measures of A and C. Therefore, its opposite side is the largest side of the triangle.
Note that a calculator can be used to verify that the cosine of an acute angle is greater than 0, that the cosine of a right angle is equal to 0, and that the cosine of an obtuse angle is less than 0.

Example

Solving Problems Using the Law of Cosines

Once Diego completed a grueling month-long shift as a lighthouse keeper, he decided to fly from San Juan to New York. After flying for 3 hours on a straight path, he felt that the pilot made a course correction, then continued to fly for about 2 more hours on a path still toward New York. On Diego's return flight, the pilot flew on a straight path, without any change in direction, from New York to San Juan.
Plane
If the plane is deflected and its average speed is 330 miles per hour, what is the distance from San Juan to New York? Round the answer to the nearest hundred miles.

Hint

Use the speed formula to calculate the distance traveled before the change in direction SD, and the distance traveled after the change DN.

Solution

First, SD and DN will be found. Then, the Law of Cosines will be used to find NS, the distance between New York and San Juan.

Finding SD and DN

The average speed is the distance traveled divided by the amount of time spent traveling.
The flight from S to D takes 3 hours and the speed of the plane is 330 miles per hour. Substitute these values into the formula and solve for SD.
Solve for SD
990=SD
SD=990
The distance covered in the first 3 hours of the flight is 990 miles. Similarly, the distance covered in the next 2 hours can be calculated.
Length SD DN
Substitution
Calculation SD=990mi DN=660mi

Finding SD

Since the plane was deflected from the first route, the measure of the angle SDN is
Now, in SDN, the lengths of two sides and the measure of their included angle are known. Therefore, the Law of Cosines can be used. Let d, s, and n be the lengths of the sides opposite D, S, and N, respectively.
Substitute 660 for s, 990 for n, and for D.
Solve for d

Approximate to nearest hundred

Since a length cannot be negative, only the principal root is considered here. Therefore, the distance from San Juan to New York is about 1600 miles.

Closure

Calculating Distances in Space Using Trigonometry

In this course, the use of the Law of Cosines in solving any type of triangle has been studied. By using this law, the challenge presented at the beginning can be solved.

Zosia knows the lengths of two sides of a triangle and the measure of their included angle. Let A, V, and D denote the vertices of the Summer Triangle.

What is the angular distance between Deneb and Vega? Round the answer to the nearest integer.

Hint

The Law of Cosines states the relationship between the side lengths of a triangle and the cosine of one of the angles.

Answer

Let a, v, and d be the lengths of the sides opposite A, V, and D, respectively. By the Law of Cosines, the following equations holds true for AVD.
Since the values of v, d, and A are given, the first equation will give the length of the third side.
Substitute these values and solve for a.
Solve for a
The angular distance between Deneb and Vega is about 24 angular units.
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