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Algebra (all content)

Course: Algebra (all content)>Unit 14

Lesson 6: Pythagorean identity

Proof of the Pythagorean trig identity

The Pythagorean identity tells us that no matter what the value of θ is, sin²θ+cos²θ is equal to 1. We can prove this identity using the Pythagorean theorem in the unit circle with x²+y²=1. Created by Sal Khan.

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• How is tan theta= cosine theta over sin theta?
• Think back algebra. Slope of a line is y/x. Tan represents that slope. Since sin θ is y and cos θ is x. So sin θ/cos θ = y/x = tan θ.
• Is there an alternative way to prove the Pythagorean trig identities (there are three of them)?
• We can use the definitions of the functions and the following property: x^2 + y^2 = r^2.
sinθ = y/r
cosθ = x/r
sin²θ + cos²θ
= (y/r)² + (x/r)²
= y²/r² + x²/r²
= (x² + y²)/r²
= r²/r²
= 1
We have proven sin²θ + cos²θ = 1. To get to the other identities, all we have to do is divide by sin²θ or cos²θ.
(sin²θ + cos²θ = 1)/cos²θ
tan²θ + 1 = sec²θ

(sin²θ + cos²θ = 1)/sin²θ
1 + cot²θ = csc²θ
• Doesn't this proof. only prove the Pythagorean Identity in the unit circle? If the radius was not one, wouldn't this not work.
• I had the same question. After a while I figured that it would still work because of how ratios work. Basically to have any other circle you would have to multiply by the same factor:

sin²Θ + cos²Θ = 1
(sin²Θ + cos²Θ)*factor = 1*factor(for different radius)

If you divide each side by the factor, you're back where you started.

I know this answer is super late, but I hope someone else can learn from it...I hope this is correct.
• At , why does he put absolute value around the sin and cos?
• Good question! Actually, I don't see why Sal would use the absolute value of sin and cos as the square of a negative integer is equal to the square of the absolute value of the same integer.

-a^2 = a^2
• What if the radius is not 1?
cos^2(theta) + sin^2(theta) = 4

Do we just divide both sides of the equation by 4?
• For the pythagorean trig identity? It works for any radius circle. Try going throught he video again but instead of x^2 + y^2 = 1, pretend that it is x^2 + y^2 = a^2 where we can insert any number for a, so it would be applicable for any circle.

The one twist is that where he has a point's coordinates be cos(theta), sin(theta), it would be a*cos(theta), a*sin(theta) since each point is found by manipulating cos(theta) = x/a and sin(theta) = y/a. let me know if you do not understand why this is.

You would not wind up with cos^2(theta) + sin^2(theta) = 4. Keep in mind x^2 + y^2 = 1 is the same as x^2 + y^2 = 1^2, because int he equation for a circle the number at the end is the radius squared.
• Is there a difference between sin²θ and (sin θ)²?

I cannot seem to understand the difference. It's also a bit confusing to put the exponent there as -1 is used to note the inverse function.
• Regarding the confusing use of superscript `-1` to indicate an inverse trig function, you can choose to use the "arc" names such as `arcsin` -- especially in your own notes, or in computer science contexts where the "arc" names predominate.

The Wikipedia article on inverse trig functions uses the "arc" notation, for example: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Notation

It's still important that you be able to use both notations, but we aren't always "stuck" with the inferior `-1` notation.
• Is sin^2(x) = sin(x)^2 and the same for cos. I am a bit confused?
• sin²(x) is the same thing as sin(x)², as is cos²(x) = cos(x)².
This notation was adopted to avoid confusion between sin(x²) and sin(x)². Why?
Often we just write sinθ rather than sin(θ), so now it is easier to see the difference between sinθ² and sin²θ.
When I first saw sin²(x) and sin²x, I preferred to write (sin(x))², but quickly got used to writing less symbols. You will too.
• I'm doing it in 8th grade, but most of my friends are doing it in 9th
• At , how are you able to find the "equation" of a circle, I just have no idea, I thought you could only graph lines that actually go somewhere?