Lesson LAW OF COSINES

Source code of 'LAW OF COSINES'

This Lesson (LAW OF COSINES) was created by by Theo(13342)

This lesson provides an overview of the Law of Cosines

REFERENCES

<a href = "http://hyperphysics.phy-astr.gsu.edu/Hbase/lcos.html" target = "_blank">http://hyperphysics.phy-astr.gsu.edu/Hbase/lcos.html</a>
<a href = "http://www.mathwarehouse.com/trigonometry/law-of-cosines-formula-examples.php" target = "_blank">http://www.mathwarehouse.com/trigonometry/law-of-cosines-formula-examples.php</a>
<a href = "http://www.mathsisfun.com/algebra/trig-cosine-law.html" target = "_blank">http://www.mathsisfun.com/algebra/trig-cosine-law.html</a>
<a href = "http://www.sparknotes.com/math/trigonometry/solvingobliquetriangles/section4.rhtml" target = "_blank">http://www.sparknotes.com/math/trigonometry/solvingobliquetriangles/section4.rhtml</a>
<a href = "http://www.clarku.edu/~djoyce/trig/oblique.html" target = "_blank">http://www.clarku.edu/~djoyce/trig/oblique.html</a>
<a href = "http://www.cliffsnotes.com/study_guide/Law-of-Cosines.topicArticleId-11658,articleId-11573.html" target = "_blank">http://www.cliffsnotes.com/study_guide/Law-of-Cosines.topicArticleId-11658,articleId-11573.html</a>
<a href = "http://www.mathopenref.com/lawofcosines.html" target = "_blank">http://www.mathopenref.com/lawofcosines.html</a>

LAW OF COSINES FORMULAS

The formulas used in the law of cosines are as follows:

{{{c^2 = a^2 + b^2 - 2*a*b*cos(C)}}}

{{{a^2 = b^2 + c^2 - 2*b*c*cos(C)}}}

{{{b^2 = a^2 + c^2 - 2*a*c*cos(B)}}}

LAW OF COSINES TRIANGLE LABELING CONVENTION

The triangles that the law of cosines applies to are labeled in the following manner:

Triangle is labeled ABC

Side a is opposite angle A.
Side b is opposite angle B.
Side c is opposite angle C.

This convention is used throughout this lesson.

LAW OF COSINES FORMULA INPUT REQUIREMENTS

In order to be able to apply the law of cosines directly, you need to be provided with the length of 2 sides and the measure of the included angle between them or the length of 3 sides.

If you are provided with the length of 2 sides and the measure of the included angle between them, you can then derive the third side.

Once you have the third side, you can then derive the other two angles.

If you are provided with the length of 3 sides, you can then derive all three angles.

LAW OF COSINES APPLICATION

The law of cosines is used on any triangle where the required inputs are provided.

The formula can be used on all triangles, including right triangles.

We will do some examples to show you how the law of cosines works.

EXAMPLE 1

You are provided with the length of two sides and the included angle between them.

You are asked to find the length of the third side.

The given sides are equal to 3 and 9 units in length.

The angle included between these two sides is given as 30 degrees.

Assign side b to be equal to 3 units in length.
Assign side c to be equal to 9 units in length.
Assign angle A to be equal to 30 degrees.

Angle A is opposite side a.

That puts angle A between sides b and c as will be seen from the sketch of your triangle.

Draw a preliminary sketch of your triangle as shown below:

Side a is opposite angle A.
Side b is opposite angle B.
Side c is opposite angle C.

Side b = 3 units in length.
Side c = 9 units in length.
Angle A is between side b and side c.

The unknown side is labeled "a" and the angle that is given is angle A.

When you are given two sides and the included angle between them, the given angle will always be opposite the unknown side.

The formula you need to use is:

{{{a^2 = b^2 + c^2 - 2*b*c*cos(A)}}}

The unknown side is �a�.
The known angle is �A�.

The known sides are �b� and �c�.

Substitute known values in this formula to get:

{{{a^2 = 3^2 + 9^2 - 2*3*9*cos(30)}}}

Simplify to get:

{{{a^2 = 9 + 81 - 54*.866025404}}}

Simplify further to get:

{{{a^2 = 90 - 46.7653718 = 43.2346282}}}

Take the square root of both sides of this equation to get:

a = 6.575304418

You now have:

a = 6.575304418
b = 3
c = 9
A = 30 degrees

Since you were asked to find the length of the third side, then you are done.

EXAMPLE 2

You are given that three sides of a triangle are 6.575304418 and 3 and 9 units in length.

You are asked to find all three angles of the triangle.

Assign these to the following sides of your triangle.

a = 6.575304418
b = 3
c = 9

Assuming you want to find angle C first, then the formula you will use will be:

{{{c^2 = a^2 + b^2 - 2*a*b*cos(C)}}}

The angle you want to find is always on the right side of the equation.   The side opposite to that angle is always on the left side of the equation.

Substitute for a,b,c to get:

{{{9^2 = 6.575304418^2 + 3^2 - 2*6.575304418*3*cos(C)}}}

Simplify this to get:

{{{81 = 43.2346282 + 9 - 39.45182651*cos(C)}}}

Combine like terms to get:

{{{81 = 52.2346282 - 39.45182651*cos(C)}}}

Subtract 52.2346282 from both sides of this equation to get:

{{{28.7653718 = -39.45182651*cos(C)}}}

Divide both sides of this equation by -39.45182651 to get:

{{{cos(C) = 28.7653718/(-39.45182651) = -.72912649}}}

Take the arc-cosine of -.72912649 to get:

C = 136.8132145 degrees.

Assume you want to find angle A.

Angle A was given as 30 degrees in example 1.

We will derive it anyway just to show you how the process works.

The formula you will use to find angle A will be:

{{{a^2 = b^2 + c^2 - 2*b*c*cos(A)}}}

The side opposite the angle is always on the left side of the equation.   The other two sides are always on the right side of the equation.

Substitute for known values in this equation to get:

{{{6.575304418^2 = 3^2 + 9^2 - 2*3*9*cos(A)}}}

Simplify this to get:

{{{43.23462819 = 9 + 81 - 54*cos(A)}}}

Combine like terms to get:

{{{43.23462819 = 90 - 54*cos(A)}}}

Add {{{54*cos(A)}}} to both sides of this equation, and subtract {{{43.23462819}}} from both sides of this equation to get:

{{{54*cos(A) = 90 - 43.23462819}}}

Combine like terms to get:

{{{54*cos(A) = 46.76537181}}}

Divide both sides of this equation by 54 to get:

{{{cos(A) = .866025404}}}

Take the arc-cosine of .866025404 to get:

A = 30 degrees.

You now have angle A and angle C.

You want to find angle B.

Since the sum of the interior angles of a triangle is always equal to 180 degrees, you can find the third angle by subtracting the sum of the two angles that you already have from 180 degrees to get:

Angle B = 180 - (30 + 136.8132145) = 180 - 166.8132145 = 13.18678547 degrees.

You now have all the measures of your triangle.

They are:

a = 6.575304418
b = 3
c = 9
A = 30 degrees
B = 13.18678545 degrees
C = 136.8132145 degrees

Since you were asked to find all 3 angles of the triangle, then you are done.

EXAMPLE 3

You are given that 3 sides of a triangle are 3,4,5 respectively.

You are asked to find all 3 angles of this triangle.

Since you are given 3 sides, you use the Law of Cosines to find the angles of this triangle.

You label the triangle ABC and you assign the sides as follows:

Side a = 3
Side b = 4
Side c = 5

Assuming you want to find angle A first, so you use the formula:

{{{a^2 = b^2 + c^2 - 2*b*c*cos(A)}}}

Substitute known values into this equation to get:

{{{3^2 = 4^2 + 5^2 - 2*4*5*cos(A)}}}

Simplify this equation to get

{{{9 = 16 + 25 - 40*cos(A)}}}

Combine like terms to get:

{{{9 = 41 - 40*cos(A)}}}

Add {{{40*cos(A)}}} to both sides of this equation and subtract 9 from both sides of this equation to get:

{{{40*cos(A) = 41 - 9}}}

Combine like terms to get:

{{{40*cos(A) = 32}}}

Divide both sides of this equation by 40 to get:

{{{cos(A) = 32/40 = .8}}}

Take the arc-cosine of .8 to get:

Angle A = 36.86989765 degrees.

Assuming you want to find angle B next, the formula you will use is:

{{{b^2 = a^2 + c^2 - 2*a*c*cos(B)}}}

Going through the same process that you used to find angle A, you will determine that:

Angle B = 53.13010235 degrees.

You find angle C by subtracting the sum of angle A and angle B from 180 degrees to get:

Angle C = 180 - (36.86989765 + 53.13010235) = 180 - 90 = 90 degrees.

You now have all the sides and angles of your triangle, so you are done.

They are:

Side a = 3
Side b = 4
Side c = 5
Angle A = 36.86989765 degrees.
Angle B = 53.13010235 degrees.
Angle C = 90 degrees.

If you had known that triangle ABC was a right triangle, you could have simply used the Pythagorean Formula.

HOW TO DETERMINE WHICH LAW OF COSINE FORMULA TO USE

If you are given 2 sides and the included angle, and you label the given sides a and b, then you will label the included angle C, and you will use the formula:

{{{c^2 = a^2 + b^2 - 2*a*b*cos(C)}}}

If you are given 2 sides and the included angle, and you label the given sides b and c, then you will label the included angle A, and you will use the formula:

{{{a^2 = b^2 + c^2 - 2*b*c*cos(A)}}}

If you are given 2 sides and the included angle, and you label the given sides a and c, then you will label the included angle B, and you will use the formula:

{{{b^2 = a^2 + c^2 - 2*a*c*cos(B)}}}

The included angle is always on the right side of the equation.

The side opposite the included angle is always on the left side of the equation.

If you are given 3 sides of the triangle, then it's your choice as to which angle you want to find first.

The side opposite the chosen angle is always on the left side of the equation.   The other two sides are always on the right side of the equation.

HOW TO DETERMINE WHETHER YOU SHOULD USE THE LAW OF SINES OR THE LAW OF COSINES OR THE PYTHAGOREAN FORMULA

The law of sines and the law of cosines both solve for the sides and angles of any triangles, including right triangles.

The use of one or the other depends on what you are given.

If you are given two sides and the included angle, then use the Law of Cosines.

If you are given three sides, then use the Law of Cosines.

If you are given two sides and an angle opposite one of those sides, then use the Law of Sines.

If you are given two angles and one side, then use the Law of Sines.

If you know that the triangle you are dealing with is a right triangle, then use the Pythagorean Formula.

The lesson on the Law of Sines can be found by clicking on the following link.

<a href = "http://www.algebra.com/algebra/homework/Trigonometry-basics/THEO-20100406.lesson" target = "_blank">LAW OF SINES</a>

The lesson on derivation of law of sines and law of cosines can be found by clicking on the following link:

<a href = "http://www.algebra.com/algebra/homework/Trigonometry-basics/THEO-2012-01-16.lesson" target = "_blank">http://www.algebra.com/algebra/homework/Trigonometry-basics/THEO-2012-01-16.lesson</a>