 So an important idea in mathematics and in life is identity It's important to distinguish between expressions identities and equations an Expression is a valid sequence of variables constants and operators So 3x plus 7 is an expression This horrible thing is an expression Cosine is not an expression because this is not a valid sequence It says to take the cosine of something, but doesn't specify the something Five divided by is not an expression because it's not a valid sequence It says divide five by something, but doesn't identify the something But if we do write something there, it does become an expression In this particular case because we're dividing by zero and we're not allowed to do that this expression is meaningless But it's still an expression An equation is a statement claiming the equality of two expressions. So some equations might be And finally an identity is an equation that is always true regardless of the value of the variables So if I were to take x plus 7 and square it I always get x squared plus 14x plus 49 and if I were to take x plus 2 It's always the same as 2 plus x The wonderful thing about identities identities are wonderful things is that their equations we can use in any problem a Related notion of some importance is this idea of a paradox a paradox is an equation That is never true regardless of the value of the variables. So x plus 2 equals 5 plus x is never true Square root 3x plus 8 equals negative 5 is never true because square root always indicates a non-negative number and C-can theta equals zero also a paradox because there is no value for which this is true The wonderful thing about identities identities are wonderful things, but there is more than one of them the good news is most of them are Unimportant and in fact, there's only a couple that are really important and most of them center around what's called the Pythagorean identity So here's a quick recap of trigonometry on the unit circle So remember if theta is nanglin standard position It's terminal side will intersect the unit circle at a point where the y-coordinate is sign The x-coordinate is cosine and the tangent is y over x So suppose we have a unit circle centered at the origin the equation of the circle is x squared plus y squared equals 1 since x equals cosine theta and y equals sine theta and Equals means replaceable we can replace and get what's known as the Pythagorean identity For any angle theta sine squared theta plus cosine squared theta equals 1 And it's worth keeping in mind that because this is always true Regardless of the angle theta this is an identity and remember an identity is an equation You can always use in any problem For example, suppose sine of theta is 3 fifths and theta is a second quadrant angle. Let's find the cosine of theta So we'll start by drawing a unit circle centered at the origin with an angle theta in the second quadrant Since we can always use an identity in any problem We know that sine squared theta plus cosine squared theta must equal 1 We know the value of sine of theta. It's given to us So it can substitute in and solve for cosine And we have two possible values of cosine theta plus or minus four fifths Now remember that the cosine of theta is going to be our x-coordinate and Since we do a picture we see that the x-coordinate of our point must be negative And so since theta is an angle in the second quadrant cosine of theta must be negative And that means the cosine of theta must be negative four fifths Now you really only need sine squared theta plus cosine squared theta equals 1 but It is helpful to have some other identities and the other identity that's really important comes from our observation that the Tangent of theta is the sine of theta divided by the cosine of theta Since an identity is an equation we can use in any problem We can start with sine squared theta plus cosine squared theta equals 1 if we divide through by cosine squared theta Our first term can be rewritten as sine theta over cosine theta squared The second term is just one and the third term can be rewritten as 1 over cosine theta squared And here's where knowing those definitions is useful tangent of theta is sine theta over cosine theta So we can replace one of our cosine theta is the same as secant theta so we can replace and We get another identity also known as a Pythagorean identity For any angle theta tangent squared theta plus 1 is equal to secant squared theta So suppose tangent theta equals 12 over 5 and theta is an angle of the third quadrant find the values of sine and cosine Since we solved every other problem in trigonometry by drawing a picture we should Draw a picture We'll draw a unit circle centered at the origin with angle theta in the third quadrant Third quadrant Now remember the sine and cosine of the angle are going to be the y and the x coordinates and Since the angle is in the third quadrant We have x less than zero and y less than zero and since x is Cosine of theta and y is the sine of theta it follows that cosine of theta must be less than zero and sine of theta must be less than zero From our Pythagorean identity sine squared theta plus cosine squared theta equals 1 or Tangent squared theta plus 1 equals secant squared theta Since we know the tangent of theta is 12 fifths we can substitute solve for secant theta and Since we're looking for sine and cosine we found neither of them Remember that by definition secant theta is one of her cosine theta So since secant theta is one of her cosine theta then cosine theta is one of her secant theta And so that means that cosine of theta is plus or minus five-thirteenths But there can be only one cosine of theta must be either five-thirteenths or negative five-thirteenths So which one is it fortunately? We've drawn a picture Remember the x-coordinate corresponds to the cosine of the angle and since theta is an angle of the third quadrant x must be negative so cosine of theta must be negative five-thirteenths To find sine of theta we can do a number of different things But probably the easiest is to use our relationship tangent theta equals sine theta over cosine theta So sine theta is equal to tangent theta times cosine theta And we know the value of tangent theta and the value of cosine theta so we can calculate the value of sine theta