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Proof of the Pythagorean trig identity

CCSS.Math: ,

Video transcript

so let's review the unit circle definition of trig functions a little bit right over here I've drawn a unit circle and when we say unit circle we're talking about a circle with radius 1 so for example this point right over here is the point 1 comma 0 X is equal to 1 Y is 0 this point is the point 0 comma 1 this is the point negative 1 comma 0 and this is the point 0 comma negative 1 that the radius over here the distance from the center of the circle which is at the origin to any point in the circle or any point on the circle I should say this radius is equal to 1 so the unit circle definition of trig functions leverages this unit circle that's why it's called the unit circle definition and we saw that if we define an angle as the kind of bottom side of the angle being along the positive x axis and then the other side of that angle thinking about where it intersects the unit circle so let's say that this is the angle theta we define we define sine of theta and cosine of theta or cosine theta and sine of theta as the X&Y coordinates of this point at which this this side of the angle that's the side that is not the positive x-axis where that intersects the unit circle so for example this point right over here if we would call this the x coordinate of this point so this value right over here we would call that cosine of theta we would call that cosine of theta and the y coordinate of that point which is this point right over here we would call that we would call that sine of theta and in previous videos on the unit circle we talked about why this is really just a natural extension of the sohcahtoa definition what's useful is it starts to work for negative angles it works for angles it even works for 90-degree angles it works for angles more than 90 degrees that works for angles less than 90 degrees so it's really really really useful but what I want to do is leverage what we already know about the unit circle definition of trig functions to help prove the Pythagorean identity it almost falls out of the fact that this point right over here is on a circle a circle of radius guess one so what is the equation of a circle with radius one centered at the origin well the equation of that is x squared x squared we have other videos where we really proved this using the distance formula which is really just an application of the Pythagorean theorem the equation of a circle of a unit circle centered at the origin is x squared plus y squared plus y squared is equal to is equal to 1 is equal to the radius squared this distance right over here is equal to 1 well we've already said that we're defining cosine of theta as the x-coordinate of this point and we're defining sine of theta is the y-coordinate of this point and this point is sitting on the circle it has to satisfy this relationship right over here so that means well if we're defining cosine of theta to be the X to be this x value sine of theta to be the Y value and it has to satisfy this relationship that means that cosine of theta squared plus sine squared of theta plus sine squared of theta needs to be equal to needs to be equal to 1 or sine squared theta plus cosine squared of theta needs to be equal to 1 and that's just from the point this is the X cosine theta is the x-coordinate sine theta is the y-coordinate they have to satisfy this relationship which defines a circle so cosine squared theta plus sine squared theta is 1 now this is called as we've seen in other videos this is called the Pythagorean Pythagorean identity and you say why is that useful well using this if you know sine of theta you can figure out what cosine of theta is going to be or vice versa and if you know one of cosine of theta and then you can say you know cosine theta then you use this to figure out sine of theta then you can figure out tangent of theta because tangent of theta is just sine over cosine but if you're a little bit confused as to why this is called the Pythagorean identity well it really just falls out of where the even the equation of a circle even came from if we look at this point right over here if we look at this point right over here which we're saying is the whereas cosine theta and the y coordinate is sine of theta what is the distance between that point and the origin well to think about that we can construct a right triangle so this distance right over here this distance right over here and so that we can deal with any quadrant I'll make it the absolute value of the cosine theta is this distance right over here and this distance right over here is the absolute value of the sine of theta the absolute value of the sine of theta and I'm obviously don't have to take the absolute value for this first quadrant here but if I went into the other quadrants and I were to set up a similar right triangle then the absolute value is at play and so what do we know from the Pythagorean theorem this is a right triangle here the hypotenuse has length one so we know that this expression squared the absolute value of cosine of theta squared plus this expression squared which is this length plus the absolute value of the sine of theta squared needs to be equal to the length of the hypotenuse squared which is the same thing which is going to be equal to 1 squared or we could say week this is the same thing if you're going to square something the sine if it's negative it's going to be negative times a negative so it's just going to be positive so this is going to be the same thing as saying that the cosine squared theta plus sine squared theta plus sine squared theta is equal is equal to 1 so once again or I guess this is why it's called the Pythagorean identity and actually that's even where the equation of a circle comes from it comes straight out of the Pythagorean theorem where your hypotenuse is a well it has length 1