 We are now going to discuss properties of trigonometric functions The first set of properties we will discuss would be the periodic properties and what this tells us is how often Trig functions repeat themselves We've already kind of discussed previously that every 360 degrees every trig function repeats itself and what this periodic property is telling us is that If you take an angle and you add 2 pi to it or 360 degrees Then it doesn't change the value of the trig expression at all that goes for sine cosine cosecant and secant Tangent we can do a little bit better than that Tangent actually repeats itself every pi radians or every 180 degrees tangent and cotangent The two trig functions that are reciprocals of each other repeat themselves every 180 degrees or every pi radians We are now going to use these periodic properties to evaluate trig expressions In our first example, I have sine of 17 pi over 4 which is 4 point something pi So I need to get this down To an angle that is between 0 and 2 pi because then I can locate the reference angle if needed and evaluate the trig expression So 17 pi over 4 I'm going to take away 2 pi because that is how often the sine function repeats itself every 2 pi radians Common denominator, so you'll take 17 pi over 4 and you will take away 8 pi over 4 I had to convert 2 pi into 8 pi over 4 This actually is going to give me 9 pi over 4 Which is still greater than 2 pi. I need to be between 0 and 2 pi So well, how about we take 9 pi over 4 now and we take away 2 pi again Which we already know is equal to 8 pi over 4. This is actually going to give me the angle pi over 4 So sine of 17 pi over 4 is actually the exact same thing as saying sine of pi over 4 And remember what we discussed previously pi over 4. Well, that's just 45 degrees So all I'm doing is finding sine of 45 degrees the angle is already in quadrant one So I don't have to worry about finding a reference angle and worrying about is my answer positive or negative all trig functions are positive in quadrant one So I'll draw our triangle draw some nice triangle here It has to be a 45 45 90 triangle Label the sides 1 1 square root of 2 remember square root of 2 always goes across from the 90 degree angle Sine of 45 degrees is opposite over hypotenuse So go to any 45 degree angle the side opposite is 1 the hypotenuse is square root of 2 So you have 1 over square root of 2 you have to rationalize because you do have a radical in the denominator So multiply by square root of 2 over itself And you have square root of 2 over 2 as your final answer here So all I did here was I took away Increments of 2 pi until I got to a final angle measure that was between 0 and 2 pi 0 and 360 degrees another example Cotangent of 405 degrees So cotangent how often does cotangent repeat itself? Well, this one's every 180 degrees So I'm going to take away 180 degrees from 405 and I'm actually going to get 225 degrees We can actually take away 180 degrees again So remember cotangent repeats itself every 180 degrees so now This is going to give us cotangent of 45 degrees Now you just need to evaluate cotangent of 45 degrees Now keep in mind that if you were to keep 225 degrees, which is a quadrant 3 angle you would have had to have found the reference angle By taking away 180 degrees Which would have then given you the reference angle of 45 degrees But it doesn't matter if you're in quadrant 1 or if you're in quadrant 3 we're going to have a positive answer either way So cotangent of 45 degrees Let's make our 45 45 90 triangle So 1 1 Square to 2 cotangent Remember cotangent is the reciprocal of tan Tangent is opposite over adjacent So cotangent has got to be adjacent over opposite Not that it really matters because adjacent side is 1 opposite is 1 so you get 1 over 1 which is 1 Positive 1 is your answer so We previously discussed in the last video the signs of trigonometric functions based on what quadrant the original angle is in So remember the quadrant of an angle Determines if the value of the trig function or trig expression is positive or negative So the quadrant of the angle determines the sign of that trig expression Remember the phrase all students take calculus all means in quadrant 1 all of the trig functions are positive Students quadrant 2 that means sign and its inverse cosecant are positive Quadrant 3 tangent and its inverse cotangent are positive Everything else is negative and quadrant 4 cosine and its inverse secant are both positive Everything else is negative. So let's use this to answer a few questions We're going to name the quadrant where each angle theta lies So I'm going to draw my little visual over here all students take calculus All right, so they tell me that Find out what quadrant theta is and given that tan theta is positive and Cosine theta is negative greater than zero is positive less than zero is negative All right, so where is tangent positive? Well, guess what that would have to be in quadrant 1 because everything's positive in quadrant 1 and quadrant 3 Then we look here and say well where's cosine negative? Well, cosine is positive in one and cosine is positive in four so it has to be negative in two and three so okay The only caught quadrant where tangent is positive yet cosine is negative is Looking here is quadrant 3 So theta is in Quadrant three Now let's look where secant is positive, but sine is negative secant is positive in Well secants with cosines so four and one Where you could say one four That is where secant's positive now sine is negative Where's sine negative not quadrant one? It's positive there not quadrant two. It's positive there It had to be quadrants three and four so three and four and So what quadrant is secant positive, but sine is negative? Quadrant four So our angle theta in this case would be in Four last exercise on this page. Where's cosecant positive? Cosecant and sine are related to each other. They're reciprocals. Well, if sine's positive in two so is cosecant So cosecant's positive and quadrant one quadrant two So we have one comma two Where's cotangent negative? cotangent is negative and Not one not in three. It's positive there. It had to be two and four What quadrant do they have in common where it is our theta lie? Theta is in Check it out quadrant two Theta is in quadrant two here. All right, so how about some more exercises here? These are a little bit different They give me an angle theta and they're like well if sine theta is three over five And cosine theta is negative four fifths find tan of theta and secant of theta Well, how about we draw a picture one of our favorite problem-solving strategies? I'm gonna draw me a triangle with theta Sign of theta History fifths sign of theta history over five opposite history Hypotenuse is five So opposite history hypotenuse is five Then cosine of theta is four fifths. So adjacent is four Hypotenuse is five. I'm ignoring the negative sign for now So Now I have myself a triangle and I have me an angle theta and I can actually find tangent and secant of that angle So let's first find tangent of theta opposite over adjacent opposite over adjacent three over four Now let's find secant of theta the reciprocal of secant is Cosine cosine is adjacent over hypotenuse Meaning secant would have to be hypotenuse over adjacent. So five over four So five over four now we haven't determined if we're gonna stay positive if these or if these should be negative That's what we're going to do now What all depends on what angle or what quadrant theta was in so looking here I have to answer the question where is sign Positive but Cosine negative so similar to what we did previously sign is positive and one or two cosine is negative and Not one not four two or three What quadrant do they have in common? Well, that would be quadrant two So theta is in two If that's the case then in quadrant two, what is tangent? Is it positive or negative? Tangent is negative If we're in quadrant two, what is secant? Secant is also negative the only trig functions that are positive and quadrant two would be sine and cosecant Everything else is negative. So go to each of our answers here and put the negative sign in front of them And those are your answers Secant's negative five over four tangent is negative three-fourths Since that one was so much fun Let's do another one Start off with a picture. So I start with a picture. I have my theta Cosine is six over seven. I'm ignoring the negative for now. This is adjacent This is hypotenuse adjacent is six hypotenuse of seven. So go to your angle theta adjacent is six hypotenuse of seven So what goes here? How do I figure that out? Well, the right angle might give it away, but yes Pythagorean theorem is what we want to do So you'll have six squared Plus y squared equals seven squared You'll have 36 plus y squared equals 49 Remember our goal here is to find y. So I'm going to take away 36 from both sides So you get y squared equals Believe that's going to give us y squared equals 13 Y squared equals 13 Well, just take the square root of both sides And you'll get y equals the square root of 13 So I now know this other side here. This other leg is square root of 13 The ultimate goal will be finding sine of theta and cotangent of theta So make me make a little bit of a note over here sine theta Cotangent of theta All right, so sine of theta opposite over hypotenuse square root of 13 over 7 I don't know if it's positive or negative yet. We'll find out in a second cotangent of theta cotangent is Adjacent over opposite So six squared six over square root of 13 We'll rationalize this by multiplying by the square root of 13 over itself This gives you six square roots of 13 over 13 Now are these answers positive or negative? Well, what quadrant is theta in? Theta is in if you want to write the degrees up here 180 degrees in 270 What quadrant is between 180 degrees in 270? I believe that would have to be quadrant 3 So if we're in quadrant 3 that means only tangent and cotangent are positive so cotangent is positive Which would mean sine? Would have to be negative So in my answers over here sine. I need to throw the negative sign in front and Then cotangent can stay positive. So I have negative square root of 13 over 7 six square roots of 13 over 13 Now there's some other cool properties these trig functions have and this falls back onto the symmetry of what these look like graphically So you might remember even an odd functions from a previous class just not when talking about trig functions So an even function means It has y-axis symmetry So function is symmetric with respect to the y-axis if it's even and odd function means it has Symmetry with respect to the origin also known as rotational symmetry And what this means is if you have a negative sign inside of a function You're able to pull it out. That means that the function is odd sign cosecant Tangent and cotangent are all odd functions meaning if you see a negative sign inside the function You're allowed to pull it out front So pretty cool property and that's because once again these functions graphs are symmetric with respect to the origin They have rotational symmetry Better yet there the cosine and secant are called even functions They're symmetric with respect to the y-axis and what this means is that they can simply just They eat the negative signs. It doesn't matter if you have a negative on the inside You can remove it and they're still equal to each other So these negative signs Can be removed and it won't change the value of the trig function at all Pretty neat and this is going to make our life easier as we work out a few more examples Use even odd properties to find the value of the following So the question is if there's a negative sign on the inside Do we pull it out front or does it just disappear and that goes to whatever function you're dealing with Cosine is an even function Cosine and secant are the only two even functions. They eat negative signs So this is the same thing as saying cosine of pi over four and we are professionals at this at this point because I Can make my 45 45 90 triangle? label my sides Cosine adjacent over hypotenuse So I have one over square root of two Rationalize multiply by square root of two over itself It looks like you'll get square root of two over two for your answer here So these even odd properties are actually really helpful for us Because literally cosine and secant you can just remove the negative sign next example tangent of negative 60 degrees So we have to think does tangent eat negative signs or does it bring them out front and you have to remember tangent is odd It's an odd function So you can actually take this negative sign and bring it out front So you'll have negative tangent of 60 degrees and we know how to find tangent of 60 degrees So I guess we'll draw 30 60 90 triangle label your side Relationships here and I'm dealing with tangent of 60 tangent of 60 so opposite over adjacent Opposite over adjacent square root of three over one So the answer is just negative Square root of three All right, so we'll do one more example together Looks like I'm now dealing with cosecant So the question is what is cosecant even or odd can I pull out the negative sign or does it just disappear and cosecant is actually an odd function So let's bring out that negative sign So now have negative cosecant of 7 pi over 6 This is an angle that is a little bit more than pi. This angle is in quadrant 2 so We'll have a little bit of work to do here. All right, so make a little note. That's a 7 pi over 6 is in quadrant 2 so I Need to find a reference angle of this and so I have to use our formulas from the previous video the reference angle if An angle is in sorry. This should actually be quadrant 3 if an angle is in quadrant 3 To find the reference angle you just subtract pi Which can be written as 6 pi over 6 my reference angle is pi over 6 so In essence, I need to find cosecant of pi Over 6 Which by the way, remember that is that is 30 degrees there Guess we'll make a 30 60 90 triangle All right cosecant is the reciprocal sign sign is opposite over hypotenuse So this should be hypotenuse over opposite. So go to your 30 degree angle hypotenuse over opposite 2 over 1 Which is 2? All right, so the deal is I was originally in quadrant 3. What is cosecant in quadrant 3? cosecant is negative In quadrant 3 because only tangent and cotangent are positive So that means I have to make this 2 negative and put it up here in the top cosecant of 7 pi over 6 The negative sign carries from the front, but cosecant of 7 pi over 6 is actually negative 2 So negative 2 over 1 or negative 2 however you like to write it regardless this ends up being just 2 in the end Because the negative out front and the negative from the trig function do Create a positive a positive 2 So I hope you've enjoyed looking at these different properties that trig functions have There are also a few identities that Trig functions have and we've kind of discussed these a little bit The first is to reciprocal identities that cosecant is the reciprocal of sign Meaning cosecant is 1 over sine theta secant is the reciprocal of cosine So secant theta is 1 over cosine theta Cotangent is the reciprocal of tangent meaning cotangent theta is 1 over tangent theta. They're reciprocals of each other Then quotient identity identities you have tan of theta is equal to sine over cosine and then cotangent is Cosine over sine these are some really good identities to remember So find the value of each expression These are actually a little bit fun to do to be careful because they can be a little bit addicting sometimes So we know That secant Secant here is the reciprocal of cosine. So secant of 74 is actually 1 over cosine of 74 degrees You multiply something by its reciprocal the answer is just 1 Everything cancels out so you can cross cancel diagonally here and the answer is 1 Next cosecant of pi over 6 times sine of pi over 6 Well, we already know that cosecant is 1 over sine We keep the angle one side the same So it looks like we're taking Sine and multiplying by its reciprocal Once again, you have this nice cancellation here and the answer is 1 All right, what about part c this one might be a little bit more involved in what we did previously So there's a few things you can do here well a lot of things you can do here The first thing is I'm going to try and get the same angle one side each of these trig functions So I have cotangent here. I have cosine over sine I'm thinking that maybe we can combine this eventually and turn this into cotangent So first off, let me take cotangent and subtract 180 degrees from it because remember cotangent repeats itself every 180 degrees now The deal is I'm trying to get 39 and all of these Well, what can I do here to this lovely sign? Well, remember sign repeats itself every 360 degrees. So how about we add 360 degrees to that inside angle? We almost have positive 39 degrees in every single angle Well, what can we do if negative sines inside sine functions? We can pull them out front So because sine is odd Once again because sine is odd. I can pull that negative sine out front Of the sine function now Keep the negative sine cosine over sine. It's the same angle on the inside cosine over sine is actually cotangent So negative cotangent of 39 degrees plus cotangent 39 degrees is actually a big fat zero All right, and that's all we have for you today So I hope you enjoyed and that you appreciate the properties and identities that we have. Thank you for watching