now there are some fundamental trigonometric identities that you need to know what we're going to talk about first is the reciprocal identity now when you think of the word reciprocal what do you think that means for example what is the reciprocal of seven the reciprocal of s is 1/ 7 so now how does this relate to trick so what is the reciprocal of s the reciprocal of sin Theta is cosecant s is 1 / cose and cose is 1/ s now this identity here is the one that you're going to use more often if you have sign for the most part you won't need to change it to cosecant but if you have cosecant a lot of times you will change it back into sign the reciprocal identity of secant Theta is 1 / cosine that's another one that you need to know and if I was you hopefully you have a notebook you want to write down these formulas because you will use them a lot throughout your Trigon um your trick course I lost the word there now coent is one divid tangent and tangent is also 1 / C tangent this one is more commonly used and cosine if secant is 1 over cosine cosine is 1 over secant now you're not going to use that often but the ones that are highlighted in boxes secant cosecant and cotangent those are the ones that I would commit to memory you should know those three so those are the reciprocal identities that you need to be familiar with with so let's say if we wish to calculate secant of Pi / 3 how can we do so how can we use reciprocal identities to figure this out well secant we know it's 1 cosine so we got to find the value of cosine Pi 3 and use the unit circle so pi over 3 is equal to 60° and at an angle of 60° this corresponds to a point on a unit circle that's 12 comma < tk3 / 2 so cosine is associated with the x value therefore cosine pi over 3 is 12 so 1 /2 is 2 and so that's the value of secant pi over3 now let's try some other examples so what what is cose of 4 Pi / 3 feel free to pause the video and try that example so first let's draw the circle or a portion of the in circle and for pi 3 is in quadrant 3 now the reference angle of 4 Pi 3 is pi 3 and we know the point that corresponds to Pi 3 it's 12 comma < tk32 so make sure you know the values in the first quadrant because you can use that to find the values in other quadrants so 4 pi over 3 will have the same numbers but different signs in quadrant 3 both X and Y are negative so this corresponds to the point -2 < tk3 over two so now let's use the reciprocal identity cose is 1 / so we got to find the value of sin 4i 3 sin is associated with the Y value so sin 4i 3 is < tk3 / 2 1 / < tk3 over2 you just got to flip the fraction that's going to be equal to -2 over < tk3 now at this point we need to rationalize the denominator so let's multiply the top and the Bottom by root3 so the final answer is -2 < tk3 / 3 this becomes theare otk of 9 which is 3 So This Is The Answer go ahead and calculate secant of 330° so go ahead and try that now what I like to do is find a reference angle 330 is in Quadrant 4 so this is an angle of 330 the reference angle has to be 30 it's 360 minus 30 so we know the angle for 30° or the values for 30 at 30° this corresponds to the point < tk3 over 2 comma 12 30 is the same as pi over 6 now 330 is going to have those same values the only difference is y is negative but X is positive so at 330 this corresponds to a point of3 over2 comma 12 and that's why it's helpful to be able to find a reference angle because once you find a reference angle you could find a point on a unit circle and then from that point you could find the other point based on what quadrant you're in you just got to adjust the signs and that's why the reference angle is so useful so now let's use the reciprocal identity secant is 1/ cosine so we need to find a value of cosine 330 now cosine 330 is going to be the x value that's < tk3 over 2 so this is what we have 1 / < tk3 over 2 is the same as 2 over < tk3 and if you want to see y view it this way this fraction is equal to 1 over 1 / < tk3 over2 so you can rewrite it as 1 1 / < tk32 and then using the Keep Change Flip principle you keep the first fraction as same change division to multiplication and flip the second fraction so in the end it becomes 2 over3 and that's why we're able to Simply flip the fraction at this point we need to rationalize so the final answer just like before is going to be positive 2 < tk3 over3 instead of a negative answer so that's cant 330 now for those of you who want access to my complete online trigonometry course here's where you could find it uh go to emi.com and then in the search box you could just search for trigonometry and you can see my course is basically the one with the black uh background and then here is it I'm still adding more lectures but here's what I have so far um introduction into angles drawn angles converting degrees into radians uh linear speed angle speed problems Arc Length uh information on the unit circle how to evaluate trig functions using the unit circle right triangle trigonometry things like SOA even you could have video quizzes as well solving work problems like angle of elevation problems and then you have the next section graphing s cosine functions secant tangent inverse trig functions pretty much all the common stuff that you'll see in a typical uh trigonometry of course even solving uh barings verifying trigonometric identities uh summon difference formulas double angle half angle and some other things too and as I mentioned before I'm going to add some other things as well so feel free to check it out when you get a chance and uh let's continue back to the video now let's try cosecant -225 go ahead and try that problem so what is cosecant -225 well let's plot -225 so this is 90- 180 and -225 so therefore it's in quadrant 2 now notice the difference between uh this value which is - 180 going this way and this is -225 so you can see the reference angle has to be 45 also if you add add 360 to -225 this will give you POS 135 so this is 135 and 135 minus positive 180 this is positive 180 that will also give you 45 so you can write this as -225 or as 135 it's up to you those two are co-terminal angles now the reference angle of 135 is 45 which is in quadrant 1 now for 45 you need to know what the X and Y values are and it's < tk2 over2 comma < tk2 over2 so now we could find the values for 135 or -225 that corresponds to this point x is going to be negative but y will remain positive so now that we have the terminal point that corresponds to -225 we can evaluate cosecant of -225 cosecant is one over S and S of -225 is the same as s 135 co-terminal angles have the same trigonometric value s is associated with the yalue so that's going to be positive < tk2 / 2 if we flip it it's going to be 2/ < tk2 anytime something is under one flip the fraction now we need to rationalize it so this becomes 2 < tk2 and the < TK of 2 * the < TK of 2 is theot of 4 which itself is two and these will cancel so the final answer is positive square < tk2 that's cose of -225 what is secant of -7 pi/ 6 we have to try negative angle in radians so you can Master this topic so let's begin by graphing it so this is piun over 6 and then here is5 piun over 6 and then -7 piun / 6 is in quadrant two now7 piun 6 is in the same location as 5 piun 6 what you can do is add 2 pi to get the coterminal angle 2 pi is the same as 12 Pi 6 if you multiply 2 pi over 1 by 6 over 6 this will allow you to get common denominators so this corresponds to 5 Pi 6 so those are called terminal angles so they have the same trig value now the reference angle for -7 piun over 6 or 5 piun 6 is simply < / 6 now the point that corresponds to Pi / 6 is < tk3 over 2 comma 12 pi over 6 is 180 / 6 which is 30° so now we can find a point that corresponds to7 Pi 6 in quadrant 2 x is negative but Y is positive so we only have to change the x coordinate now secant is 1/ cosine so now we could find the value of cosine -7 piun / 6 and cosine is associated with the x value so this is going to be 1 over < tk3 / 2 which is -2 over < tk3 and then if we rationalize it we're going to get this familiar answer again -2 < tk3 over3 so after a while you begin to notice some common answers and if you see an answer that's not common and you have an angle that's in a unit circle you can quickly identify the wrong answers go ahead and evaluate these functions secant of 0° and secant of 90 now to do so we need to be familiar with some points and that is the angle zero and 90 and the XY values that occur at those points so at 0 deg we have the point 1 comma 0 x is 1 but Y is always Z on the x axis on the Y AIS X is z but Y is going to be one so using those points what is the value of secant of 0° now secant is 1 / cosine so we got to find cosine of 0° and cosine is the x value so cosine of 0 is 1 and 1 / 1 is 1 so secant of 0 Dees simply has a value of one now what about secant of 90 secant 90 is 1 / cosine of 90° and cosine of 90 we need to use the x value again cosine of 90 is zero so when you get a situation like this what is 1id 0 and what is 0 ID 1 1 / 0 if you ever see it it is undefined that's what you need to write for your answer 0 / one is simply zero so you can have a zero on the top of a fraction but not on the bottom if you do have it on the bottom your answer is undefined so secant 90 simply does not exist it's an undefined value try these evaluate cosecant of 180 and also cosecant of T70 so let's try this again so here's 180 and here is 270 so at 270 we have the point 01 X is always zero on the Y AIS and at 180 Y is Zer on the x-axis this is - 1 comma 0 cosecant is 1 / s so we need to use the Y value for s s of 180 is 0 so this is going to be 1 / 0 therefore cose of 18 80 is undefined now cosecant of 270 that's 1 / sin 270 and sin 270 once again is the yalue it's 1 so 1 /1 is1 so that's the value of cosecant 270 it's negative 1 for