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### Course: Precalculus>Unit 2

Lesson 5: Law of cosines

# Solving for an angle with the law of cosines

Sal is given a triangle with all side lengths but no angle measure, and he finds one of the angle measures using the law of cosines. Created by Sal Khan.

## Want to join the conversation?

• Is the inverse of cosine (cos^-1) the same as arc cosine (arccos)?
• Yes, arccos and the inverse of cosine (cos^-1) are the same thing.
• In Sal's previous video, he said the formula was a^2=b^2+c^2-2ab cos Theta.
why is he now using c^2 instead of a^2?
• The variables are reversible. You can put a^2, b^2, or c^2 on the left side of the equation, you just have to fix the other side of the equation to not contain the variable used on the left side.
• At what is the actual value of cosine? How can we calculate the cosine of a number if we don't have a calculator?
• You won't be asked to do that. The actual computation is far too difficult to do by hand in a reasonable about of time.

Instead, you will be expected to memorize the sines and cosines of some special angles. Other than that, either you will be allowed to use a calculator or you'll be given the values.

The actual computation for cosine (angles expressed with radians, not degrees):
cos x = ½ [ e^(-i*x) + e^(i*x)]
You will not be expected to do this until an advanced course in calculus.
• At why didn't Sal just take the -6000 and add it to the other side, thus isolating theta ?
• Well the term -6000 is together with the cosine of theta. For example if 10=5x you can't subtract 5 from each side to get x.
• Around Mr. Khan moved the (cos) from one side of the problem to the other. When it moves wouldn't it have to be divided over? Why is he still multiplying cos-1 to the rest of the problem when he should be dividing it? I'm confused. Is there a secret rule I'm missing?
Is it that when (cos) is moved over it only becomes (cos-1) or something?
• Your last sentence is correct. The cos⁻¹(x) is the inverse function to cosine(x). You could say it "undoes" the cosine function, so whereas cosine takes an angle and returns a ratio, cos⁻¹ takes a ratio and returns an angle.

You could regard what Sal did as taking cos⁻¹ of both sides, so we'd have
cos⁻¹(cos(θ)) = cos⁻¹((19/20)
So in the LHS we take the cosine of theta, and then take the inverse cosine, which is just theta, so we have
θ = cos⁻¹((19/20).

Also be aware that there are alternative names for the inverse trigonometric functions: cos⁻¹ is also called arcosine, sin⁻¹ is arcsine, and tan⁻¹ is arctangent.
• Why is their no law or rule for tangent?
• There is a Law of Tangents!

The Law of Tangents has been around since at least the 13th century, when Persian mathematician Nasir al-Din al-Tusi wrote about it in his book, Treatise on the Quadrilateral.

Law of Tangents:
(a-b)/(a+b)
= [tan ½(α - β)]/[tan ½(α + β)]

Hope this helps!
• At , do we always try to simplify the fraction?
• A calculator will do that for you. If you put 15/24 into your calculator and press enter, you will get 5/8, which is the simplified form of 15/24.
• Why did Sal do 400-6100 when he could have done 6100-6000?
• 6100 and 6000 are not like terms because of the variable with the 6000. So let x = cos(theta). You have 400=6100-6000x which is a two step equation. So to solve, you subtract and divide.
• At , when Sal subtracts 6100 from both sides, why can't he just do this:

400 = 6100 - 6000 x cos theta
400 = 100 x cos theta
400 / 100 = cos theta
4 = cos theta