Today's soundtrack is Shania Twain: Up! (Green Version), the country-styled version of her classic album. Every one of the nineteen songs on the album is a modern classic; each one is so infectiously catchy, with sing-along lyrics and head-bobbing music.
As we learned in the lesson about the sine law (part 1, part 2), if we know either the measure of two angles and the length of one side or the length of two sides and the measure of one non-contained angle, we can use the sine law to solve the triangle. But what if we know the lengths of two sides and the measure of one contained angle, or if we know the lengths of all three sides but no angles?
Enter the cosine law: c² = a² + b² - 2ab cos C.
Remember the Pythagorean Theorem? It tells us that in a right triangle, a² + b² = c². The reason that it looks so similar to the cosine law is that the Pythagorean Theorem is actually a "special case of the Cosine Law" (Pearson's Pre-Calculus 11, p. 492). The reason it works is that cos 90° = 0, meaning that if we let C represent our right angle, our equation becomes c² = a² + b² - 2ab cos 0, which simplifies to the classic c² = a² + b².
As I alluded to above, we can use the cosine law to solve triangles in two situations:
1) We know the lengths of two sides and the measure of a contained angle
2) We know the lengths of all three sides, but do not know the measures of any angles
Note that in either situation, there is only one possible solution. We do not need to worry about ambiguous cases when dealing with the cosine law.
Let's go through the steps of each.
How to use the cosine law to find the length of any side
If we know the lengths of the two sides and the angle where they connect (the "enclosed" angle), the third side has only one possible length.
Write out the cosine law
c² = a² + b² - 2ab cos C
Substitute the known measurements into the equation. The unknown side will be represented by c or an equivalent variable, regardless of whether it is the hypotenuse.
Calculate the value of the right side of the equation
Take the square root of both sides of the equation to remove the square from the left side of the equation
Label the triangle accordingly
How to use the cosine law to find the measure of an angle
If we know the lengths of all three sides of a triangle, there is only one combination of angles that they can have; it is only possible to create one specific triangle out of those sides.
Write out the cosine law
c² = a² + b² - 2ab cos C
Substitute the known measurements into the equation
Collect like terms
Isolate cos C
Divide both sides of the equation by the coefficient preceding cos C
Isolate C
Move cos from the left side of the equation to the right side of the equation, which converts cos to cos⁻¹
With C isolated on the left side, multiply the right side of the equation by cos⁻¹
Label the triangle accordingly
How to use the cosine law and the sine law in conjunction
If we are asked to solve a triangle where we know two sides and the enclosed angle, we can use the cosine law to find the length of the third side, then we can use the sine law to find the two missing angles.
Use the cosine law to find the length of the missing side
Write out the cosine law
c² = a² + b² - 2ab cos C
Substitute the known measurements into the equation.
Calculate the value of the right side of the equation
Take the square root of both sides of the equation to remove the square from the left side of the equation
Label the triangle accordingly
Use the sine law to find one of the missing angles (θ)
Write out the sine law equation with the angles in the numerator position
Substitute the three known values into the equation
Solve for sin θ
Multiply the known side that's under the unknown angle by the known angle
Divide the result by the length of the other known side
Isolate θ by moving sin to the other side of the equation, turning it into sin⁻¹
Multiply sin⁻¹ by the value beside it
Label the triangle accordingly
Use the triangle sum theorem to determine the measure of the final angle
180 - A - B = C
Label the triangle with the final angle
That's all for tonight! Next time, I'll be working on the assignment portion of this lesson and sharing some examples of my work.