- Intro to the Pythagorean trig identity
- Sine & cosine of complementary angles
- Using complementary angles
- Relate ratios in right triangles
- Trig word problem: complementary angles
- Trig challenge problem: trig values & side ratios
- Trig ratios of special triangles
Sal introduces and proves the identity (sinθ)^2+(cosθ)^2=1, which arises from the Pythagorean theorem! Created by Sal Khan.
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- Are arccos and secant the same?(16 votes)
- 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.(48 votes)
- At2:08, he says "sin^2(theta)" why didn't he say "sin(theta)^2"?(22 votes)
- 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".(26 votes)
- 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 ?(7 votes)
- 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. :)(41 votes)
- Sal writes sin²Θ + cos²Θ = 1, shouldn't it be (sinΘ)² + (cosΘ)² = 1?(10 votes)
- 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).(20 votes)
- Does the theta have to always bee in that exact place or can it be any unknown angle?(5 votes)
- At3:16, 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?(5 votes)
- At3:16right? 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.(3 votes)
- Does this identity apply to non-right triangles as well?(4 votes)
- 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)
- Is sin^2(theta) equal to the sin(theta), and then all of that squared? Or is it theta squared, and then the sine of that? I'm guessing it's the former, but just making sure. . .(3 votes)
- 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.(2 votes)
- 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
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(4 votes)
So we've got a right triangle drawn over here where this base's length is a, the height here is b, and the length of the hypotenuse is c. And we already know when we see something like this, we know from the Pythagorean theorem, the relationship between a, b, and c, we know there's a squared plus b squared is going to be equal to the hypotenuse squared, is going to be equal to c squared. What I want to do in this video is explore how we can relate trig functions to, essentially, the Pythagorean theorem. And to do that, let's pick one of these non-right angles. So let's pick this angle right over here as theta, and let's just think about this what the sine of theta is and what the cosine of theta is, and see if we can mess with them a little bit to somehow leverage the Pythagorean theorem. So before we do that, let's just write down sohcahtoa just so we remember the definitions of these trig functions. So sine is opposite over hypotenuse. Cah, cosine is adjacent over hypotenuse. And toa, tan is opposite over adjacent, we won't be using tan, at least in this video. So let's think about sine of theta. I will do it, I'll do it in this blue color. So sine of theta is what? It is opposite over hypotenuse, so it is equal to the length of b or it is equal to b-- b is the length-- b over the length of the hypotenuse, which is c. Now what is cosine of theta? Well, the adjacent side, the side of this angle that is not the hypotenuse, it has length a. So it's the length of the adjacent side over the length of the hypotenuse. Now how could I relate these things? Well it seems like, if I square sine of theta, then I'm going to have sine squared theta is equal to b squared over c squared, and cosine squared theta is going to be a squared over c squared. Seems like I might be able to add them to get something that's pretty close to the Pythagorean theorem here. So let's try that out. So sine squared theta is equal to b squared over c squared. I just squared both sides. Cosine squared theta is equal to a squared over c squared. So what's this sum? What's sine squared theta plus cosine squared theta? Is going to be equal to what? Sine squared theta is b squared over c squared, plus a squared over c squared, which is going to be equal to-- Well we have a common denominator of c squared. And the numerator, we have b squared plus a squared. Now, what is b squared plus a squared? Well, we have it right over here, Pythagorean theorem tells us, b squared plus a squared or a squared plus b squared is going to be equal to c squared. So this numerator simplifies to c squared. And the whole expression is c squared over c squared, which is just equal to 1. So using the sohcahtoa definition, in a future video, we'll use the unit circle definition. But you see just using the units, just even using the sohcahtoa definition of our trig functions, we see probably the most important of all the trig identities. That the sine squared theta, sine squared of an angle, plus the cosine squared of that same angle-- I'm introducing orange unnecessarily-- is going to be equal to 1. Now you might probably be saying, OK Sal, that's kind of cool, but what's the big deal about this? Why should I care about this? Well, the big deal is now you give me the sine of an angle and I can solve this equation for the cosine of that angle, or vice versa. So this is actually a pretty powerful, powerful thing. And this is also part of the motivation even for the unit circle definition of trig functions.