You know how to use trigonometric ratios to find missing sides in right triangles. You also know how to use the Law of Sines to find missing sides in non-right triangles given certain information. What about the triangle below? Can you solve for ?
The Pythagorean Theorem relates the three sides of a right triangle.
When you increase the measure of , the length of the opposite side increases. There is a new relationship between the sides of the triangle.
Similarly, when you decrease the measure of , the length of the opposite side decreases. There is a new relationship between the sides of the triangle.
The Law of Cosines takes these relationships one step further. It uses the measure of angle to provide an equation that relates the three sides of the triangle with angle .
Law of Cosines:
In the past, you used the Pythagorean Theorem to find missing sides in right triangles. With the Law of Cosines, you can find missing sides and angles in any type of triangle. In the examples you will learn how to prove the Law of Cosines.
Given with lengths and , find in terms of and . Then, use the Pythagorean Theorem twice to come up with two equations that relate with the sides of .
Solution: and , so . You can use the Pythagorean Theorem with both and .
Use the triangle and equations from Example A to derive the Law of Cosines.
Solution: From Example A, you have two equations that you can solve for .
Set the new equations equal to each other, expand, and simplify:
You can now consider in terms of . Within , is adjacent to . is the hypotenuse. This is a cosine relationship.
Substitute this equation for into the earlier equation:
Here you have derived the Law of Cosines for acute angles . Just like the sine ratio, the cosine ratio is actually a function that takes any angle measure as an input (not just angles between and ). One property of the cosine function is that the cosine of supplementary angles will always be opposites. In other words, . In the practice exercises, you will use this fact to show that the Law of Cosines works even if is a right or obtuse angle. Note: You will study the cosine function in much more detail in future courses!
Use the Law of Cosines to solve for .
Solution: 15 and 20 are the values for and , and is the unknown.
Concept Problem Revisited
Notice that in this triangle, the included angle is obtuse. As was explained in Example B, the Law of Cosines works for acute, right, and obtuse angles. The names of the sides do not match , and from the Law of Cosines exactly. What is important is that the angle you choose for is the included angle of the two sides you choose for and . Side is opposite .
In this triangle, the sides 18 and 14 have an included angle of . is opposite the angle. Use the Law of Cosines to relate these 4 values.
The cosine (cos) of an angle within a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
The trigonometric ratios are sine, cosine, and tangent.
Trigonometry is the study of triangles.
, or “theta”, is a Greek letter. In geometry, it is often used as a variable to represent an unknown angle measure.
The Law of Cosines states that for any triangle with sides , , and and opposite from side , .
Solve for or .
1. The Law of Cosines states that . Remember that the names of your sides might not match , and exactly. What is important is that is the included angle of sides and . Side is opposite .
In this triangle, the sides 11 and 6 have an included angle of . is opposite the angle. Use the Law of Cosines to relate these 4 values.
2. In this triangle, the sides 8 and 15 have an included angle of . is opposite the angle. Use the Law of Cosines to relate these 4 values.
3. In this triangle, the sides 9 and 13 have an included angle of . 11 is opposite the angle . Use the Law of Cosines to relate these 4 values.
Note that you can use the Law of Cosines to solve for missing angles as well as missing sides.
1. What is the Law of Cosines?
2. When using the Law of Cosines, how do you decide on the values for , and ?
3. Show that the Law of Cosines is identical to the Pythagorean Theorem when is a right angle.
Use the Law of Cosines to solve for or .
5. The following triangle is isosceles. Solve for by first figuring out the measure of .
8. Attempt to solve for . Explain why you get two solutions.
In problems 10-12, you will investigate the relationship between the cosine of supplementary angles.
10. Find and . What do you notice?
11. Find and . What do you notice?
12. Make a conjecture based on the last two problems. How are the cosine of supplementary angles related? Note: You will prove this conjecture in future courses!
In problems 13-16, you will prove that the Law of Cosines works even if is an obtuse angle.
13. Use the Pythagorean Theorem to write an equation relating , and . Then, use the Pythagorean Theorem again to write an equation relating , and .
14. Use algebra and your work from #13 to show that the following equation is true: .
15. Find an equation that relates and and . Show that your equation is equivalent to . Solve for and substitute this quantity into the equation from #14.
16. The cosine of supplementary angles are opposites. Substitute for . Show that the result is the Law of Cosines.