11 ## THE LAW OF COSINESWE USE THE LAW OF COSINES AND THE LAW OF SINES to solve triangles that are not right-angled. Such triangles are called oblique triangles. The Law of Cosines is used much more widely than the Law of Sines. Specifically, Thus if we know sides
(The Law of Cosines is a extension of the Pythagorean theorem, because if θ were a right angle, we would have
Example 1. In triangle DEF, side
Problem 1. In the oblique triangle ABC, find side To see the answer, pass your mouse over the colored area.
= 5
= 25 + 2 − 10
= 25 + 2 − 10, ( = 17.
Problem 2. In the oblique triangle PQR, find side
= 25 + 100 − 100(.970), from the Table. = 125 − 97 = 28.
Example 2. In Example 1, we found that Use the Law of Sines to complete the solution of triangle DEF. That is, find angles E and F.
Therefore, on inspecting the Table for the angle whose sine is closest to .944, Angle F 71°. And therefore,
And so using the Laws of Sines and Cosines, we have completely solved the triangle. The Law of Cosines is also valid when the included angle is obtuse. But in that case, the cosine is negative. See Topic 16. Proof of the Law of Cosines Let ABC be a triangle with sides
(The trigonometric functions are defined in terms of a right-angled triangle. Therefore it is only with the aid of right-angled triangles that we can prove anything) Draw BD perpendicular to CA, separating triangle ABC into the two right triangles BDC, BDA. BD is the height Call CD Also, since
then
Now, in the right triangle BDC, according to the Pythagorean theorem,
so that
In the right triangle BDA,
For
Finally, for
That is,
This is what we wanted to prove. In the same way, we could prove that
and
This is the Law of Cosines. Please make a donation to keep TheMathPage online. Copyright © 2022 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |