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### Course: Precalculus (Eureka Math/EngageNY)>Unit 4

Lesson 4: Topic B: Trigonometry and triangles

# Solving for a side with the law of cosines

Learn how to use the law of cosines to find the missing side length of a triangle when given two side lengths and the contained angle measure. Created by Sal Khan.

## Want to join the conversation?

• Maybe I'm just not quite getting this, but why not just use the Pythagorean Theorem? It comes out to 15, right?
• You can ONLY use the Pythagorean Theorem when dealing with a right triangle. The law of cosines allows us to find angle (or side length) measurements for triangles other than right triangles. The third side in the example given would ONLY = 15 if the angle between the two sides was 90 degrees. In the example in the video, the angle between the two sides is NOT 90 degrees; it's 87. As such, that opposite side length isn't 15; it's 14.6.
• What's the difference between a theorem and a law?
• Good question!
In science at least, here is the difference between a theory/theorem and a law:
A theory is an explanation for a natural occurrence. It tells the "why" about something, but it has not necessarily been proven.
A law, on the other hand, states a fact- something that always happens. It tells the "what" without explaining why, and it should always be true.
• Is there a Law of Tangent?
• Yes, you can find it on Wikipedia. But it's equivalent to the Law of Sines, so it's not really useful.
• in the equation,a^2=b^2+c^2-2bc cos(theta),does a have to be the longest side
or can it be any side
• "a" in the law of cosines is the side opposite of the angle theta, so it can be of any length.
• In what situation do you use the law of cosines?
• You could use it if you know SSS and want to find an angle, or if you know SAS and want to find the remaining side.
• WHy are they assigning this in preschool
• Lol, I am assigned as the teacher for my brothers and sometimes for fun I would assign them tasks that they couldn't do. XD That was a few years back.
• At couldn't you just use the Pythagorean Theorem?
• No, because it's not a right triangle (or, at the very least, we can't prove it to be a right triangle).
• At just under one minute into the video, Sal discussed; if we draw sides b and closer, the angle between them will be small, and so will the length opposite it... Why did he talk about this in this video at that moment?
What point was he trying to get across?
• Well, if sides b and c move closer together, or their angle decreases, side a will become shorter and shorter. Thus, we must figure out the angle of Θ before we attempt to figure out side a's length, as the angle must be a constant, otherwise a will not be a constant.