now let's talk about the pythagorean identities there's three of them that you need to be aware of the first one is the most common sine squared of theta plus cosine squared theta is equal to one that's the first one here are the other two one plus tangent squared is equal to secant squared and 1 plus cotangent squared is equal to cosecant squared so those are the pythagorean identities but let's focus on the first one and how we can use it so let's say sine theta is equal to 4 over 5. and let's say the angle theta is between 0 and 90. using this information and this is in degrees by the way what is the value of cosine theta how can we find the value of cosine so using the pythagorean identity sine squared plus cosine squared is one if we know the value of sine we could find the value of cosine sine is four over five so now let's calculate the value of cosine four squared is sixteen five squared is twenty five and one we can write it as twenty five divided by twenty five now the next thing that we could do is we could subtract both sides by 16 over 25 so cosine squared theta is 25 over 25 minus 16 over 25 25 minus 16 is 9 so we have 9 over 25 and now we need to take the square root of both sides the square root of 9 is 3 the square root of 25 is 5. now it can be plus or minus three over five now which sign should we choose should we choose positive three divided by five or negative three divided by five to answer that question we need to look at the angles theta is in quadrant one it's between zero and ninety degrees and in quadrant one cosine is positive so the answer that we're gonna have to choose is positive three over five so we need to use the quadrant to find the final sign let's try another example let's say cosine is equal to 8 over 17. and let's say the angle theta is between three pi over two and two pi so based on this what is the value of sine so let's use the same formula sine squared plus cosine squared is equal to one so we need to find the value of sine cosine is eight over 17. 8 squared is 64 and 17 squared is 289 and one let's rewrite it as 289 divided by 289 any number divided by itself is one so we got to subtract 289 by 64. so sine squared is equal to 225 divided by 289 now let's take the square root of both sides so sine theta is going to be the square root of 225 is 15 and we know the square root of 289 is 17. so now which sign should we use positive or negative so if the angle is between 3 pi over 2 and 2 pi what quadrant is it located in now keep in mind 3 pi over 2 is equivalent to 270. 2 pi is 360. so it's located in quadrant 4. and in quadrant 4 sine is negative but cosine is positive so therefore sine is going to be negative 15 over 17 in this particular example and so that's the answer let's try one more example let's say that sine theta is equal to two over five and tangent theta is less than zero what is the value of cosine theta so before we use the formula let's find out what quadrant the angle is located in so first we know that sine is positive sine is positive in quadrants one and four now tangent is less than zero tangent is positive in quadrants one and three tangent is y over x in quadrant one x and y are both positive so if you divide a positive number by a positive number it's going to be positive in quadrant three x and y are both negative when you divide two negative numbers you'll get a positive value in quadrants two and four tangent is negative now we know tangent is less than zero so it's less than zero or negative in quadrants two and four but sine has to be positive so both statements are true and quadrant four sine is positive but tangent is negative now cosine cosine is positive in quadrants one and four since our angle is in quadrant four we know our final answer has to be positive so just keep that in mind now let's finish it so let's start with the pythagorean identity sine is two over five two squared is equal to four and five squared is twenty-five now just like before we're going to write one in terms of this number the denominator so 1 is equal to 25 divided by 25 and 25 minus 4 is 21. so now at this point let's take the square root of both sides so cosine theta is going to equal plus or minus square root 21 and the square root of 25 is 5. we can't simplify the square root of 21. now we're going to choose the positive answer so our final answer is positive square root 21 over five since cosine since theta is in quadrant four cosine has to be positive