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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 and

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 find and 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 angular units and 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 and 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 with sides of length and which are respectively opposite the angles with measures and

triangle with angles and sides labeled

The following equations hold true with regard to

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 to its opposite side

acute triangle and its height
By the definition of an altitude, both and are right triangles. By applying the Pythagorean Theorem to and two equations can be obtained. The binomial in Equation I can be expanded.
Notice that Equation II says that is equal to Therefore, the expanded form of Equation I can be rewritten by using the Substitution Property of Equality.
Now, the term in this equation can be written using the cosine ratio.
acute angle of the triangle marker

In the cosine of is the ratio of to Finally, can be substituted for into 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 with side lengths of and respectively opposite the angles with measures and such that is greater than

obtuse triangle

The altitude of the triangle is the perpendicular segment from to the extension of the base Let be the endpoint of this segment and be the distance from to

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

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 meters and 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 and into the equation. Then, solve for
Solve for
Note that only the principal root was considered because a side length cannot be negative. Therefore, the distance between the trees is about 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 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

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

Evaluate right-hand side
The angle measures about
Triangle KLM

Finding

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
Solve for

The measure of is about
Triangle KLM

Finding

By the Triangle Angle Sum Theorem, the sum of the interior angles of a triangle is Therefore, all measures of 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 all three side lengths and the measure of the angle at are given. Examine how the length of changes as the measure of varies.

Calculate the sum of the squares of and Compare it with the square of when 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 in which all three sides lengths and the measure of the angle at are known. Let and be the lengths of the sides opposite and respectively. In the following applet, the values of and are shown. Move the slider to change the measure of
The table below shows the equation for the Law of Cosines when is an acute, right, and obtuse angle. In the table it is also shown the relationship between and for these values, and conclusions are made.
Relationship between and Conclusion
If is acute, not too many conclusions can be made. The opposite side to can be the largest side, the shortest side, or none.
The cosine of is In this case, the Law of Cosines becomes the Pythagorean Theorem. This means that the opposite side to is the largest side of the triangle.
If is obtuse, its measure is greater than the measures of and 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 that the cosine of a right angle is equal to and that the cosine of an obtuse angle is less than

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 hours on a straight path, he felt that the pilot made a course correction, then continued to fly for about 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 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 and the distance traveled after the change

Solution

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

Finding and

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

Finding

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

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 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 and 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 and be the lengths of the sides opposite and respectively. By the Law of Cosines, the following equations holds true for Since the values of and are given, the first equation will give the length of the third side.

Substitute these values and solve for
Solve for
The angular distance between Deneb and Vega is about angular units.
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