Mathematics Methods Year 11: Trigonometry I Test

Hinzugefügt: 2026-05-20

[00:00:01]okay we'll do another trig one test question one use the unit circle below to determine each of the following values now guys we should know that in a unit circle everyone should know this that always um I to draw this so let's say this is your um axis and this is your unit circle we should all know that all angles are always measured clockwise from this way they always measured clock sorry anticlockwise from the positive xaxis and they'll always have a corresponding X and a yv value we should all know that the S of the angle is always the Y value COS of the angle is always the x value and the tan of the angle is always the is sin Theta / cos Theta which means it is always the Y value divide by the x value so that's the basics that we should know about unit circle we can use a unit circle we can use a circle ccle of radius 1 to help us find the S and COS of any angle as long as the angle is anticlockwise from the positive xais so the first one cos 56 so we just have to go 56 this way and when you go 56 this way it gives us this angle here it hits the circle at this coordinate and cos is the x value so the answer is simply cos 56 is C all right it is the x value there which is C next one s 156 now that's interesting cuz this 156 is quite a way through so 156 is like here somewhere now what you meant to do is notice that there is some sort of symmetry if you do 156 guys 156 is somewhere around about here but what you meant to realize that if this here is 156 then as per symmetry cuz angles in a straight line add up to 180 that would make this here um 24 4° so what you realize that hey if that's 24° it's completely symmetrical with this which means that where would this hit the um Circle well it hits it at the same y value as this the only thing difference is the x value so basically this point here isga a and still positive B and we want the S value and remember guys s is always the Y value so what's the Y value it hits at B that's one way of doing it I'm pretty sure some students would preferred to do this they would have preferred to say that sign 156° that means that we are going all the way here that means the reference angle is 24° and what is s in that quadrant we should all by now know that all of them are positive here only s positive only tan positive and only cos positive so a lot of people would have just realized that okay we don't need to do s of 156 we can just do s of 24 and realize that the reference angle is 24° and that s is positive in that quadrant okay and that would have given them b as well and lucky last tan 336 so why don't we do it using our reference formula way so where is 336 guys 336 if I draw it completely separately 336 is like an actual actual angle of this it is an actual actual angle of all the way here which is 336° which means the reference angle is that which is um 24° So Tan 336 means the reference angle is 24° and what is Tan in this quadrant remember it's all positive s positive tan positive cos positive so it is negative in this quadrant so it is negative tan 24 and we just discussed how you find tan tan is s of 24 / COS of 204 now we can use the unit circle to help us work out s of 24° s of 24° is the Y value at 24° the Y value there is B and the x value there is a so this would be sin 24 is B whereas cos 24 is a and therefore your answer is minus B on a perfect next question convert your angle it's calc free convert your angle from 135° to radians we all should know that to go from degrees to radians you have to multiply by piun on 180 so all I have to do is do 135 * < on 180 and that's quite convenient guys because 135 on 180 you can actually simplify that to 3/4 so that is 3 Pi on 4 all right next question is the other way around they want you to go from radians to degrees and we know that if you want to go from radians to degrees you don't Times by P and 180 you divide by P and 180 AK Times by 180 on Pi so without a calculator I need to do 7 piun on 6 * 180 / piun all right Pi Pi just crosses out uh 6 and 180 becomes 30 and 30 * 7 is 210° so 210° next evaluate s 60 and tan30 we need to know our exact values sin 60 is < tk3 on 2 and tan 30 I usually write Ro < tk3 on 3 I usually write guys tan3 is < tk3 on3 cuz it's rational but I specifically wrote one on root3 because I realized that by writing one on root3 the root3 Roo t3s will cross out so the answer you're left with is half okay the ones that you need to know guys that have multiple values or like multiple ways of writing it is for example sin 45 or cos 45 the rational way of writing s 45 or cos 45 is < tk2 on 2 but be aware that it can be written as 1 on < tk2 the other one is the one that we just discussed the tan 30 yes the rational version is < tk3 on3 but it can also be written as 1 on < tk3 so please be aware that these are the ones where you can express it in like multiple different ways one with a rational denominator one without a rational denominator okay all right cos 2 pi on three you either do it in look there are two ways to do it it's your choice so students that are not so strong will have to first convert the 2 pi and 3 into de so they'll have to do 2 pi and 3 and they'll have to do times by 180 on pi and that gives us pi and Pi crosses out uh 180 on 360 that gives you 120° so student that is not strong will first convert that to 120 and then what they'll do once they've converted it to 120 what they are now able to do is realize that okay that means that the actual actual angle is one 20 and if the is that what I did before yep the actual actual angle is 120 and if the actual angle is 120 it means the reference angle is 60 so then they'll be writing cos 60 then they'll REM they'll remember a s TC only s is positive here but this says cos so therefore they'll put negative in front of it and then knowing you're exactly values that's neg half so that's how a weaker student would do it they would first convert it to degrees if your intention is to do year 12 methods we need to be comfortable with radians straight away someone comfortable with radian straight away realize that okay that's zero and that's zero radians and that's Pi radians so 2 piun and 3 means 2/3 of the way 2 piun and 3 means 2/3 of the way so that's 2 pi on 3 which means this is 1/3 of the way back which is pi on three so a strong student doesn't need to do that a strong student can immediately just do cos 2 pi on 3 they realize the reference angle is pi on 3 yet negative and they're immediately able to write half okay so up to you how quickly you get used to radians but that's the two ways the question could have been done next question circular Pizza is cut in 12 equal pieces all right 12 equal pieces all right so guys if it's cut into 12 equal pieces what is the angle subtended at the center of each of them remember we don't do 360 we do 2 pi you should all remember that we're no longer doing 360° we're going to do stuff in radians which is 2 pi so Theta is 2 Pi / 12 so each angle is pi on 6 radians and guys what are the three formulas that we need to know if angles are in radians the three formulas that we need to know that if the angle is radians is length of an arc is R Theta area of a segment is half r² Theta and area of a sorry that's not segment that's a sector this is the formula for area of a sector and area of a segment is half R 2 braet Theta T sin Theta there is another formula that you can use the other formula is special cuz for these three the angle must be in radians Theta must be in radians however the formula for length of a cord does not require the angle to be in radians the formula for length of a cord is 2 S 2 R sin Theta on 2 okay the angle does not have to be in radians this one Theta is either radians or degrees radians or degrees I hope everyone knows where that formula for length of a chord came from so if you think about it there's your chord uh and we want to know how long it is now if I was to connect it from here to here I hope everyone realizes I've created an isos triangle cuz that's the two radiuses of a circle all right and we've obviously cut this angle in half so if this angle used to be Theta but if I cut it in half each bit of the angle is just Theta on 2 and we can realize that's cut in half so it's 90° and then we can call this a remember a is half the chord so what we can do is write sign Theta on 2 equal opposite over hypotenuse a is R sin Theta 2 which means the length of the chord is 2 a which is 2 R sin thet on 2 that is a proof behind the length of the chord formula anyway back to this we now know that each angle is pi and six radians because an entire 2 pi was split into 12 pieces and we just discussed the formula for length of a cord is R Theta so it says the arc length of each Pizza is four the radius is unknown and the angle is obviously Pi on 6 we just have to bring Pi on 6 to the other side so R is Just 4 * 6 on Pi so the exact exact radius guys is 24 on Pi CM that is the exact radius of the circle easy all righty next question write down the value of the gradient now guys we really should know that the formula for the gradient of a line is tan Theta except Theta guys is the anticlockwise angle made with the xaxis the anticlockwise angle between the X AIS and the line so what I'm trying to say is if you want to find out the gradient of this line obviously guys there are other formulas for gradient rise over run AK Y2 take y1 on x2 take X X1 is another formula but in terms of angles that is a Formula so if I wanted to find the gradient of this line guys I need this angle right here all right I need that angle right there not this one here so what would this angle here be it would be 180 if you're going to do it in degrees or if you're going to do it in radians Pi this angle here is pi take Theta because angles in a straight line add up to Pi and then that's Pi take Theta so the value of the gradient here is quite simply tan Pi T Theta okay tan Pi take Theta I'll also accept anybody that wrote if anybody wrote tan of 180° you have to write the degree sign I'll accept that and I'll also accept guys there are a few kids that would have actually have used done something interesting they would have actually done and not everybody knows this but I'm just going to write it on the side So Tan of 180 I'll just do it in a different color cuz very few kids would have done this is tan is s over cos right so sin 180 take Pi on cos 180 take pi and you really should know guys that s of 180 minus an angle is the exact same as s of the angle and COS of I don't know I'm writing Pi in both spots uh even over here I don't know I write Pi I meant tan 180 T Theta okay so s of 180 take Theta is the same as s of theta and COS of 180 take Theta is cos Theta so if anybody in the class wrote - tan Theta I would have also accepted it that would have been an equally acceptable answer okay all this this or this would have been correct and these are important identities for you to know it's important for you to know that s of 180 I'm just going to write down a few important identities that you should know there the year 11 method student at this stage identities that you should know one of them is this one here one of them is knowing that sin 2 Theta + cos² Theta is 1 and what that just means guys it just means sin Theta o^ 2+ cos Theta all 2 = 1 so that's one identity you should know the other identity you should know is that tan Theta is sin Theta over cos Theta another identity that you should know at least as a year 11 method student are the following ones here s of 90 take away an angle and remember guys another way writing 90 is pi on 2 but just to make it easy I'll just do everything in degrees so it's easier for you to memorize it s of 90 take an angle is cos Theta whereas COS of 90 take an angle is tan Theta another one you should know s of 180 take Theta this is the one that we just did Sin Theta cos 180 take Theta is cos Theta uh sorry NE cos Theta and tan 180 T Theta is tan Theta so these are some important identities and a few other ones that are really handy s of theta is s of positive Theta oh sorry sorry sorry sin of theta is sin Theta COS of theta is cos Theta tan Theta is tan Theta great the these are some identities that you should definitely definitely know okay you can prove them using the unit circle all these identities can be proved using the unit circle that's not the point of this video but at least knowing the identities you should know that is important so then you could have very easily gu if you saw tan 180 take Theta you could have straight away just said oh tan Theta without having to prove why it is that okay next question Peter and Steven are kayaking from a boy Peter is 430 m away on a bearing of 113° he's 430 M away on a bearing of 13° so first uh let's just draw a boy in the middle somewhere from a boy Peter's 430 M away on a bearing of3 so that's somewhere over here and this angle here never label the entire 113 it's just messy if you write 113 label the like acute angle it makes with one of the compass points so 90 + another 23 is 113 and is obviously 430 M away from a boy that is where Peter is all right Steven is 310 M away on a bearing of 210 so 210 would be somewhere over here and he is 210 M away okay 210 M and 210 meaning this angle here is 30 by the way guys if this is 23 what does it mean this is uh 50 67° perfect what is the distance between Peter and Steven so that's very very easy we can just realize that we can just do cosine rule cuz we know this entire angle right here is 30 + 67 is 97 there is nothing stopping us from doing some non-ri angle trigonometry and cosine rule so what we have we'll call this distance X X2 is uh that shouldn't say 210 that should say 310 X2 is 310 2 + 430 2 by the way I'm using cosine rule cuz I've got two angles two sides in the angle in between 430 2 Take 2 * 310 * 430 * COS of that angle which is 97° and therefore guys you will get to two decimal places X is equal to 55 9.90 M next question what is the bearing of Peter from Steven the bearing of Peter from Steven okay so we just need to draw ourself another Compass bearing so we can find the bearing of Peter from Steven so Peter from Steven guys the angle that we are interested in is this angle right here this is the answer whatever this angle here is is the answer cuz that is the bearing from north of Peter from Steven we just have to work out that angle so I think what's going to help us is to First obviously it's going to help us if we can work out this angle right here we'll call it f we can use sign rule to work that out so what we have is is sin 5 / the opposite which is 430 = sin 97 / by the opposite side which was X which was 55 9.90 bring the 430 to the other side and then we just have to inverse sign it 430 sin 97° divide by 55 9.90 that is 5 and that gives us 44966 you could have used cosine rule again if you wanted to guys you could have used cosine rule I would have been equally happy with anyone that did this you would have done cos 5 because you do know the two sides next to it so you could have done 310 2 + 55990 S take away the opposite side squared so the opposite side squared was 43 30 2 / this make sure you guys know both versions of cosine rule the two versions of cosine rule is the one where you have two sides the angle in between that's this version right here and the other version is when you have all three sides and you'd like to find an angle and that's the one that I'm just doing right now so 2 * 310 * 55 9.90 and you inverse CA that and you would have also got 49.6 so just two different ways of doing that question now that's not the answer guys that is not the answer that is just that angle so we want the entire Theta so then we use our knowledge of Zs and Alternate angles so if that's 30 guys if that's 30 this is also 30 this here would be also 30 and then obviously we have to do 30 + 5 to get the entire angle so 30 + 49.6 6 30 + 49.6 6 is 79.6 6° so therefore the bearing of Peter from Steven is I'll accept anyone that writes 0.7 9.66 de or anyone that just wrote it like 080 true any of those will do okay you don't have to write the T but it's just um often they do that for True bearing but it's not necessary if you write it okay perfect nice question all right question six Billy Bill and Malcolm by a plot of land all right calculate the area of the whole plot of land now we should know that that's two different triangles one and another one and the area of a triangle formula is half a b sin C that is a formula for area of a triangle so what do we have here do we have this side yes do we have this side yes do we have the angle in between yes so the area of the first triangle is very very easy so I'm just going to say area is area of first triangle plus area of second triangle first triangle is very easy we just do half times by one side Times by The Other Side Times by S of 79° now it's the next one that's tricky It's tricky because they give us all three sides but they don't give us an angle so it's up to us which angle we would like to work out it really doesn't make any difference up to us which angle we'd like to work out okay I'm just going to call hypothetically I'll call this one here my Theta okay it doesn't matter any of them just had to call Theta okay so we can use cosine rule to work out Theta cos Theta equal one side squared plus the other one next to it squared take away the opposite side squared IDE 2 * 75 * 70 that gives us once you inverse cause that an angle of 51.63 De that makes the question very very easy so now all I have to do is half this this and sign of this angle here so halftimes uh 51 sorry half * 75 * 70 * s 51.63 De and then we just add those two up okay total total area guys of the land is if you want the work I do it all in one shot in my calculator on my scientific but that's 104 2488 and the other one's 2.44 adds up to guys to the nearest so first of all it's 30929 3 m squared but it wants to the nearest meter so 393 M squar great next question consider this diagram below find to the nearest meter the length of BD they want the length of BD Okay so that's not too difficult guys so first we can use this triangle here and you got to realize that if that's 65 it means this here is 180 take that which is 100 um that there would be uh 180 take 65 which is5 and that's perfect cuz if that's 115 we're able to work out that angle there that angle there is 180 take 38 take 115 so 180 take 138 take 115 should give you 27° perfect now what do we have we technically have an angle and the opposite side as long as you have angle and opposite side then you can use sign Rule and remember we're trying to find out the length of BD so BD again for those that can't visualize so first of all there's ab and that's the angle opposite AB that's going to help help us that is the angle that is opposite ab and we have technically got um we don't want this side we want the side BD and if you want the side BD you will need the angle that is opposite this side and the angle that's opposite this side is your 38° okay it's the 38° that is opposite that so that is a rule that we're going to have to use S rule so we're going to write down s 38 uh divide by the in fact I like to keep my unknown I like to keep my unknowns on the top so the unknown is the length BD right doesn't matter which way you write it but I prefer to keep the unknown on the top so in this case the side length is unknown so I'm keeping BD on the top so BD / by S 38° is equal to uh 28 on the sign of the angle opposite that which is s of 27° okay then we just have to bring the sin 38 to the other side and put that straight into your calculator make sure you're able to swap between degrees and radians appropriately so you don't get the wrong answer so 28 sin 38 on S and they want to the nearest meter okay I'm just going to work it out exactly and then rounded 28 sin 38 IDE by sin 27 make sure I'm in degrees yes all right so 28 s 38 IDE by sin 27 3797 37.979467 L CD that's so easy guys you can just do normal sakoa you can do right angle trigonometry to find that CD so we have um we can just go cuz we have the hypotenuse now and we want the opposite side so we can just do s sin 65 is just CD on BD which we worked out we'll use the exact value of 37.979467 37.979467 so we just go and write times sin 65 straight onto our calculator times s 65 and that gives us 34.419584 and again to the nearest meter 34.419584 down the middle and you got to realize that if something's tangent it means it creates in 90° so that's so easy to work out this angle cuz again you have a right angle triangle with two of the sides given you have a right angle triangle with two of the sides given so tan theta equals opposite over hypotenuse Theta is inverse tan 12 5 but we don't want Theta we want angle BCA and BCA is 2 2 Theta angle BCA is 2 Theta which is 2 inverse tan 121 5 okay and we just go to inverse 10 [Music] 1215 and we get 134.5 6 all right I'm writing that EXT decimal place just so teachers and I haven't cheated so therefore 134.0 which is what they're saying to one decimal place easy that part was straightforward next part find the area of the Shaded region so if you think about it guys what is the Shaded region so I'd like to think the area of the Shaded region is to me it is equal to the area of this Triangle Well basically the area of two triangles that are identical so it is equal to area of 2 * by area of a triangle area of the Shaded region is 2 * by the area of the triangle minus away this sector B C A minus a a area of the sector b c a all right so the formula for area triangle now I know I usually do half a sin C but because it's a right angle triangle I can actually just do half base * height so it's 2 * half base * height and we should all know the formula for area of a sector but we need it in radians formula for a sector when it's in radians is half r^ 2 Theta the two and the half cross out so what's the base 12 the height five the radius of this circle is also five and now make sure the angle is in radians so 1 34.76 use the exact value and to get it into radians we have to times it by pi on 180 that should give us easily the Shaded area so that shaded area equals 30 what does it one at two to 1 decimal place so I'll just do it exact first so we have 60 take 25 on 2 * 134.5 * piun on 180 so 60 take 12.5 take 12.5 um Times by whatever this angle here was but that's in degrees we times that by pi on 180 and that gives us 30599 basically 30.6 okay so 30.6 however they only want it to one decimal place so that would be 30.6 CM squ easy question okay all right PC is extended find the area of the Shaded region now okay well that still can't be too bad okay find the are the Shaded region now if we think about it guys if this was we'd call this Theta hadn't we we'd called yes we'd call that Theta so if we'd call this Theta guys that meant that this angle here was 180 take Theta all right now what I want everyone to realize is what do we technically have if you think about it what we technically have is a segment is this not just a segment yep that is definitely a segment segment it's a segment of this circle and what is the angle that it creates 180 T Theta is the angle that it creates so and we have how many of those segments we have two of those segments so effectively the area of the Shaded region this time the area of the red shaded region this side is just uh two area of segments and the formula for area of a segment is um the formula for area of a segment is half r^ 2 Theta T sin Theta all right except in this par well let's just be very simple the 2 and the half cross out so it's R 2 and be very very careful Theta in radians now what do we have um by the way guys I just hope everyone realized that I know I wrote half R 2 Theta in the previous question um but remember it was actually technically the entire angle here so I I I know half R 2 Theta is the generic formula but I should have really written half r^ 2 2 Theta it's okay when I subed it in when I did sub it in I didn't Sub in Theta I did Sub in 2 Theta but we just need to know what the value of theta is inverse tan of 125 but remember we want it in radians so Theta is inverse tan of 121 5 but since we want it in radians we have to multiply that by pi on 180 and that will give it to us in radians so if I get the inverse 10an of 12 on 5 oops do inverse 10 of 1215 and then times it by pi and 180 that should give it to us um whoops put the bracket here and then times it by P and 180 176 1.76 is the angle okay so back to over here is equal to um two and a half crossed out what is the radius the radius of this circle is five still 2 and the half is already crossed out the angle is 1.76 take sign now be very very careful I know we got I know okay did we do this in radians yes but yet my calculator was in the degree setting which is fine because nowhere here have I written s or cos as long as you're not writing s or cos doesn't matter whether you're in degrees or radians but now I'm writing sign and if I'm writing sign I must must must be in radians I must must must be in radians so I need to change it to rad and we want five Theta which is the answer take sign of the answer all right and that should give it to us 1.

[00:35:55]2646 okay 1 Point uh actually I just for remembered I forgot to square the radius so it's 5 squar my bad so we want 25 times by that uh so we want 25 times by Theta take sin Theta and that's actually uh make sure I've got all of that right two sectors uh no got to be very very careful guys be very careful remember we don't want Theta again I almost made a silly mistake we want 180 take Theta or in fact we don't want 180 take Theta we want pi take Theta so I almost use the wrong angle be very careful not to make silly mistakes like that the angle that we want is pi take Theta not Theta so Pi take Theta is uh Pi take 1.76 so let's go back over here and we actually want we don't want this angle here that's Theta we want pi take away that so Pi take away that this will give us the correct answer Pi minus this actually the angle is 3.12 actually the angle is 3.12 uh if I give it a bit more exact 3.12 one okay so we got the angle which is inverse tan of 1215 then we times it by the uh yes we got the actual angle make sure I haven't made silly mistakes this time so I get inverse tan of 12 on 5 and then I times it by pi on 180 to get it into RAD and then I do 180 they take away that now make sure that's all correct Pi take inverse T of 12 5 * P 180 that is correct and then we can now put it into our formula so formula that we had is area was 2 * area of a segment which is 2 * half r^ 2 Theta take sin Theta the 2 and the half cross out the radius of this circle was definitely 5 so we have 5^ squar the angle is definitely definitely 3 3.1 21 and then take s of the angle 3.12 1 take sign 3.12 1 that's much better and that will give us the area of this minor segment okay so 5 squar I'm pretty sure that everything else is correct 5 squ is 25 uh just do it properly should be the same answer this take sign of the exact same thing yep 7751 is your area makes sense all right good question next one um okay the top of a vertical rad this is a difficult one I want everyone to pause and try and do this bys okay pause and try do it by your self from the top of a vertical Tower um m stands 35 m above the surrounding level ground from point A okay so we have a big vertical Tower and I hope everyone realize is 3D so when it's guys when it's 2D you draw your compass like this when it is 2D you definitely draw your compass like this where this is North this is South this is West this is East but when it's 3D you have to draw your compass a bit differently when it is 3D you have to draw your compass like this so that's for twodimensional and for three-dimensional you have to draw it that says West for threedimensional you have to draw it like this this make this line still flat and then like this so what this means is this is North now this is East this is up this is down this is south and this is West that's how you draw it in three dimension so when I draw my tower I'm going to draw the picture relatively big all right so there's my tower and whoops we're not drawing it's 3D so it's like this and it says the top of a vertical Tower stands 35 m above level ground so there is a tower that is 35 m above the ground so we can write 35 m all right from point A which is on the ground and due east of the base all right a is due east of the base so what we're going to do is a line here a is somewhere here due east of the base and the elevation from the top of the tower from due east of the base is 30 is 50° okay the angle of elevation from due east of the base is 50° all right from another point which is 60 M away from a Now 60 M where 60 m is like an entire circle around a we don't know where that specific point is but it says from another Point C which is 60 M away the bearing of the Mast is 24° all right from another point which is 60 M away the bearing of the MK is 24° and this is the hard part it's this bit here that is hard where they say that c which is 60 M away from a the bearing of the base of the mass is 24 if you understand that you are good if you don't understand that this question becomes difficult so what I'm able to do is this I realize that okay the bearing of the MK from this point is 24° so I'm able to realize all right and it will take a while we'll take some practice for you to realize it I'm able to realize that c is somewhere here how do I know that cuz if you think about it if I put C here and that's on the ground that would make sense cuz that would be the bearing of the Mast cuz this is what m is m is the Mast so the or Mast or vertical what it bearing base of the M yeah so my m is fine okay so what you realize that okay that's all on ground so if this is all on ground then we have to turn we have a bearing of 24° in order to be pointing towards the Mast it specifically says the bearing of the Mast from C yes the bearing of the Mast from the point C is 24° okay from another Point okay so that's 24 if you understood that the rest of this question is diff is much easier if you couldn't get that then it would have been a struggle and now you even know that the distance between C to a is 60 M we did say I remember I drew a circle I had no clue where it was I didn't know if it's that that that so turns out the distance between here to here is 60 M so I can draw a line like such connect it up and say 60 M all right and they want the angle of elevation of the top of the mass from point C so what they actually want guys what they actually want is this angle so that's 35 M I can put here they want to know what is this angle here that is the question find that angle of elevation all right I'll just do it in different color so we can see it we want to find out what is that angle of elevation off from the top of the M okay so once you got the picture it's much more straightforward but a lot of people under exam pressure or test pressure for the last question did not get the picture now they we got that look how easy it's going to be um now what information do I have I got this as 35 I got this as 50 that's plenty of information to help me work out what Ma I can do simple soaka to work out the value of Ma so I'll be doing tan of 50° tan of 50° is opposite which is 35 over adjacent which is your am and therefore it is very very straightforward am is just 35 / tan 50 and 35 / tan 50 keep it exact but 35 divid tan 50 is 2937 M so I now know that this distance here is 2.37 M all right um now the next thing that I need to do then is this I also know that this is 24 so if that's 24 according to Zs according to alternate angles this here is also 24 and if that's 24 um we know that this here is 90 so that means that we actually have this entire angle I'll just draw the 90 properly this angle here is 90 so that means that we know that this entire entire angle right here the 24 plus 90 gives us an entire angle of 114° all right now there is a very important learning point here now here's the important learning point a lot of kids will do this I don't want you guys doing it I hope you've tried this yourself I hope you've paused and tried this yourself but a lot of kids will be like thinking this they'll be thinking that okay what do I have in this triangle right here let's highlight what we have we technically have this side we technically have that not technically we do have that side and we have this angle here so a lot of kids will be thinking that oh I've got an angle in the opposite side and if I've got an angle in the opposite side I have to use sign Rule now remember I'm trying to figure out this side length because if I can figure out this side length I'm just going to call it uh X for now if I can figure out this side length I'm winning okay guys I'm winning because if I can figure out this side length then I can easily just do soaka TOA all right a lot of kids will think that I've got an angle and the opposite side if if I've got an angle in the opposite side they'll think that I have to do sign rule so they'll do this they'll do sin 114 IDE by the opposite side which is 60 equals and the only angle they're able to work out well they have this side so the only other angle that they're able to work out is this angle right here we'll call it 5 is equal to sin 5 on 2937 and then you don't have to worry about the ambiguous case cuz you already have an obtuse angle there so if you work at five five will be bring the 20 9.37 across and then inverse sign so inverse sign 2937 sin 114 / 60 and when we inverse sign that that gives us a value of 26.5 six but remember guys we don't want five we want X we want this side here so now what the kids will do is they'll tell themselves that okay let's work out this angle here let's call it Alpha so Alpha we already have one another triangle in the Triangle we have another angle in the Triangle so Alpha is just 180 take the rest 180 take 26.5 6 take away the um so we're at 180 take 26.5 6 take away 114 and that gives us an angle equal to 39.449756 they will again use sign rule to find out the opposite side so they could have done this they could have now said that the opposite side is x/ s 39.449756 X is 41.7 2 now why did I use sign rule in the first place I use sign rule in the first place cuz I had an angle in the opposite side but what else do you have you have a calculator you literally have a calculator and because you have a calculator you got to think that okay I technically have this angle let's go back to what I have forget all this that I did over here you could have actually just thought of it like this you could have actually thought of it as um you have you could just like use solver so think about it you have this angle here right so you could have actually written this statement using cosine rule you could have said that 602 = X which you don't know so x^2 plus the other side which is 29.3 7^ 2 Take 2 * x * 2937 * cos1 14 you could have actually just put that straight into solver and you'll get two answers one of which will be negative I'm sure and you'll get two possible answers so let's see what I get so we have again make sure in degrees you need to be comfortable changing into degrees and radians and not forgetting forgetting is going to be very expensive if you forget to change from degrees to radians so what we have is 60 SAR equals uh we had X2 plus the other side next to it squared which would mean 2937 take 2 * x * 2937 * COS of that angle between them which was 114 and then we solve so just put a solve in front of that uh math one solve that and look at that you get two answers only one of which is positive and what do you notice the positive answer is 41.7 2 exactly what we said before 41.7 2 how much quicker would that have been how much more faster and efficient would that have been much faster and efficient use your class pad to maximum capacity and now guys it's easy you want to work out Theta that's super super easy tan Theta is opposite over the adjacent side you just worked out inverse tan that and when you inverse tan that you get 40.0 de and that guys is the angle of elevation off the top of the mass from point C okay tricky one especially for the last question question but that's the end of that trig test well done