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### Course: Trigonometry>Unit 1

Lesson 5: Sine and cosine of complementary angles

# Intro to the Pythagorean trig identity

Sal introduces and proves the identity (sinθ)^2+(cosθ)^2=1, which arises from the Pythagorean theorem! Created by Sal Khan.

## Want to join the conversation?

• Are arccos and secant the same?
• To add to Steven's excellent answer, let note a point at which beginning student get confused.
The inverses of the trig function are indicated either with arc- or with ⁻¹. However, the inverses are NOT the reciprocal of the trig functions. The confusion come from the fact that ⁻¹ is sometimes used to indicate a reciprocal. So, the problem is an inconsistency of notation that has, unfortunately, become standard.
Thus, cos⁻¹ x = arccos x. It does NOT equal sec x or 1 / cos x
The same goes for all of the trigonometric functions and their inverses.

To make it more confusing (cos x)⁻¹ does mean sec x. So, the ⁻¹ only means the inverse function instead of the reciprocal when it is written immediately after the name of the function but before the argument.
• At , he says "sin^2(theta)" why didn't he say "sin(theta)^2"?
• sin ^2 theta is the more "formal" way to write out the equation, as it avoids the possibility of mistaking the theta as "theta^2".
• So I'm doing trig in foundations 11 and we are just reviewing grade 10. In grade 10 I completely understood but now I find it harder than ever and I looked to you for help but I don't know what "THEDA" is ?
• Theta is a Greek letter. It usually represents an unknown angle measure. So...I can literally say...I don't know what Theta is either. Oh I'm gonna die laughing. That was good right? Kidding. Yeah, Theta represents the unknown angle measure, it's like a variable. :)
• Sal writes sin²Θ + cos²Θ = 1, shouldn't it be (sinΘ)² + (cosΘ)² = 1?
• Those mean the same thing. However, it is customary to write it as sin² Θ instead of (sin Θ )² to avoid confusion with sin (Θ²).
• Does the theta have to always bee in that exact place or can it be any unknown angle?
• It can be any angle. Theta just means the angle we are evaluating
• At , it says sin^2 θ+cos^2 θ=1.
So does it mean that sin θ=(1-cos^2 θ)^1/2 and cos θ=(1-sin^2 θ)^1/2?
• At right? yes, it is true that sin θ=(1-cos^2 θ)^1/2 and cos θ=(1-sin^2 θ)^1/2, we can achieve that by simple rearrangement of the equation.
• Does this identity apply to non-right triangles as well?
• No, it doesn't. This identity uses a particular property with right triangles called the Pythagorean theorem, where the hypotenuse's length is equal to the square root of the sum of the squares of the legs of the triangle. Plus, the trigonometric ratios themselves apply to right triangles. There are things that apply to all triangles, but I don't want to explain them in this reply because it may be confusing.
(1 vote)
• At the near end of the video, Sal says that this identity is part of the motivation for discovering the unit circle definitions. Why? I don't see a connection.
• According to the Pythagorean theorem, the sum of the squares of the lengths of these two sides should equal the square of the length of the hypotenuse:

x² + y² = 1²

But because x = cosθ and y = sinθ for a point (x, y) on the unit circle, this becomes:

(cosθ)² + (sinθ)² = 1

or

cos²θ + sin²θ = 1

So the identity sin²θ + cos²θ = 1 is inherent in the definition of the sine and cosine functions in terms of the unit circle, and it provides a mathematical confirmation of the geometric relationship between the unit circle and right triangles