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Basics of trigonometry.....[mumbling]

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In this lesson we're gonna be talking about triangles aren't specifically right [Triangle teaching a class of students]

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triangles and as we're sure you know right triangle is a triangle with a

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right angle also known as a ninety degree angle and wait are you confused?

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wondering if you clicked on a geometry video by mistake well no this is

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definitely physics and physics uses a lot of trigonometry sine cosine tangent [Trigonometry record playing]

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you know all the classics just in case you haven't played around with trig

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before it's the study of right triangles now you may think well what's there to

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study three sides 90 degree angle boom got it but believe it or not

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mathematicians managed to make it a bit more complicated and why is this

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important to physics well to answer that let's take a drive say we need to get

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from home to the store and the good news is it's a straight shot just a quick two [Home and store on a google map]

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kilometer drive away and hey we're physicists that's the value of length

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kilometers well we've done plenty of work with that kind of thing so we're

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just rolling along here and just knowing we have to drive two kilometers isn't [Right triangle driving a car]

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going to get us to the store you also have to know what direction to go right

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otherwise we might just end up in the middle of nowhere so we need to know

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how long the drive is and which way we're going like if the store we're

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driving to is two kilometers to the east that's the difference between scalar and

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vector quantities well a scalar quantity is it's something

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that just has a value you might see that referred to as a magnitude but when we

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said we needed to drive two kilometers that was a scalar value a vector

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quantity has both a value aka a magnitude and a direction all right so

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two kilometers to the east is a vector quantity now we're gonna come back to [Bucket of scalars appear]

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scalars and vectors later in the course but it's important we know the basics

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here why why is that important well it's pretty obvious that vectors give us more

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information and we like information and all vectors can be broken down into

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perpendicular horizontal and vertical components using......... trigonometry

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yes! all right so let's say the magic word to open up the wonderful [Magician waving wand]

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world of trig sohcahtoa not quite the classic like Shazam

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but sohcahtoa is a little more useful in the real world before we explain what

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that word means well let's take a close look at a right triangle again the

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defining characteristic of a right triangle is that 90-degree angle the [90 degree angle in triangle appears]

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side of our triangle opposite the right angle is the hypotenuse it's the longest

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side of the shape so obviously we've got two other angles and two other sides

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well the two other angles are called the complementary angles and what are those [Complimentary angles highlighted]

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sides called well they're called the adjacent and the opposite sides but

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which is which different which well it depends on which angle were working with

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so let's just pick one of the complementary angles and we'll go with

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this one well the side of the triangle that's next to the angle and not the

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hypotenuse is our adjacent side and the side that's opposite this angle is

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well the opposite side makes sense right if we look at the other angle instead [Opposite angle highlighted]

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the adjacent and opposite sides change oh and now whichever angle we're looking

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at gets this theta symbol it's just a way to show which angle we're dealing

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with there are mathematical functions that are a part of trig yeah these

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functions relate the angles of a triangle to the lengths of its sides and

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you might be asking yourself huh yeah so let's take a second and break a triangle [Man scratching head and stopwatch appears]

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down all right what makes a triangle a

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triangle those three angles right that's why it's tri angle and if we measure

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those angles with the protractors we all keep under our pillows well we find that

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when we add up all those angles the sum is 180 degrees and that goes for every

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triangle the three interior angles always add up to 180 degrees so in a [Selection of triangles appear]

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right triangle we already know that one angle is precisely ninety degrees which

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means that the other ninety degrees that are left over will be split up between

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the complementary angles so if angle B here is thirty degrees it's buddy angle

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C here will be 60 degrees and this is the basis of trig functions in fact

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let's investigate one of these functions and we'll start with the sine function

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well the sine of an angle is the ratio of the angles opposite side and the [Sine function of angle appears]

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hypotenuse and since we're dealing with a ratio it doesn't matter if the

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triangle is ginormous or if it's tiny if the angle is the same then the ratio

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between the opposite and adjacent sides will be the same another one of

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trigonometry's' greatest-hits is the cosine

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all right the cosine is the ratio of an angles adjacent side to the hypotenuse [Cosine equation appears]

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and last but not least is the tangent that's the ratio of the angles opposite

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to its adjacent side okay one more time sine is opposite over hypotenuse cosine

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is adjacent over hypotenuse tangent is opposite over adjacent how do we

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remember this sohcahtoa and you thought we were just saying gibberish all right

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well trigonometry started with this ancient Greek dude named Pythagoras and [Pythagoras with Greek friends appear]

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the ancient Greeks took their math pretty seriously and cult even crew

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around Pythagoras not a cult as in he was cool and underground and then he hit

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it big and everyone called him a sellout a cult as in an actual religious cult

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some ancient texts even claimed that if cult members told outsiders about some [Cult member whispering to outsider]

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of the math they did well, that member would and should be killed that were a

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bit more children then.. also a lot of this stuff was known in China and

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Babylon thousands of years before Pythagoras but whatever

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Pythagoras name lives on in the Pythagorean theorem might be the most

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famous mathematical formula ever or at least close second to that e equals [Pythagoras and Einstein on stage together]

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mc-squared the Pythagorean theorem says that the

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some of the squares of the two shorter sides of a right triangle, lets call

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those A and B equals the square of the hypotenuse which we'll call C so A

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squared plus B squared equals C squared and knowing this formula means that if

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we know any two sides of a right triangle we can solve for the other side

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so if we know A and B we know that C equals the square root of a squared plus [C side of triangle square root of A and B side appears]

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B squared Pythagoras you clever dog you thanks for

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the help okay so, so far we've been talking a lot of math why don't we just

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see all this math in action...Let's say we live out in the country somewhere and

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we want to see if Granny Shmoop sent us some birthday cash or our mailbox is five [Woman walks out onto front porch]

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hundred meters to the east and five hundred meters to the north we want to

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make life a little easier and take the shortest route possible well let's start

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by drawing out what we have so far aha it's like we've got ourselves a right

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angle it also looks like two sides of a triangle and the shortest path to get

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that birthday card from Grandma? Yeah, it's gonna be the hypotenuse of this triangle

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right here so our old pal Pythagoras told us that A squared plus B squared

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equals C squared and we've got an A and a B here they're both 500 so the [Pythagoras theorem and right triangle appears]

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hypotenuse will equal the square root of the sum of 500 meters squared plus 500

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meters squared go ahead grab a calculator you don't have one already [Person using a calculator]

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and our calculator tells us that the answer is 707 point whatever a whole

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buncha numbers that we don't really need to worry about because we only need two

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around the closest meter so our little shortcut will be a 707 meters long and

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we're dealing with a vector quantity here so we need to factor in that

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direction 707 meters northeast and hopefully Grandma come through with a fat [Girl opens mail box and stacks of cash appear]

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stack of cash but hold on a second just for funsies let's figure out the angle

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we'll be creating by taking this path yes this is what we consider funsies now

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we'll be honest we could totally cheat here if you have a right triangle where

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the opposite and adjacent sides are the exact same length while the two

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complementary angles are gonna be 45 degrees each but if we just spit out

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that answer we'd never get to learn about the inverse trig functions yeah [Boy studying at his desk]

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there are more than just three trig functions sine cosine and tangent each

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have corresponding inverse functions those would be the arcsine, arccosine

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and arctangent functions you might also see them written like the

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original big three but with a negative one added in the superscript not that

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the inverse functions are actually the negative first power it's really just a [Right sided triangle talking under spotlight]

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notation convention and what do we mean when we say inverse well if sine(x)

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equals y then arcsine y equals x or to put it another way if we know the

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degrees in an angle we can use the sine function to find a value for the ratio

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between the opposite side and the hypotenuse if we have a right triangle

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with a complementary angle of 30 degrees the sine of that angle will be 0.5 also [30 degree angle of triangle appears]

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known as 1 over 2 and since sine equals opposite over hypotenuse well that means

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that the hypotenuse will be twice as long as the angles opposite side but

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what if we don't know the value of that angle well if we know how long the

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hypotenuse is and the opposite side is 30 meters for the big H here and 15 for

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its little pal oh well we can do that division and find a ratio of 1 over 2

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also known as 1/2 yeah we can pop that number into the arcsine function to see

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how many degrees are in this angle and what do you know while the arcsine

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function tells us the angle is 30 degrees so sine uses the degrees in an [30 degree angle highlighted]

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angle to find how the opposite side and hypotenuse are related while its inverse

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function arcsine uses the ratio of the opposite side and the hypotenuse to find

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how many degrees are in the angle so let's find the value of this angle by

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starting with its tangent sohcahtoa reminds us that the tangent is the

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opposite over the adjacent and in this case both are 500 meters meaning our

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tangent is 1 we can get the value for the angle by applying the inverse [Tangent angle formula appears]

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function to each side of the equation now on the left hand side the inverse

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tangent and tangent will cancel each other out

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making the value of the angle equal to the arctangent of 1 plug that into your

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calculator making sure you're using the inverse tangent and that it's set for [Boy using calculator]

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degrees and we'll find out the value of the angle is yeah drumroll.......

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45-degrees kind of anticlimactic since we already said that's what it would be

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but still starting to see how all these triangles are gonna help us influence [Right triangles helping to lift mans boxes]

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well that wasn't super heavy on the physics side so look at one more problem

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remember our pendulum experiment well it's back baby we're not gonna just

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forget about our little pendulum buddy so let's say we've got a pendulum that's [Man puts pendulum on door frame]

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half a meter long and we want to know what the height of the weight would be

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if we pull the pendulum to a 30-degree angle well because when we pull the

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weight to this side it'll be higher up right here's a diagram it looks pretty

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triangular doesn't it we've got L for the total length of the string and H for the

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difference in height and we want to figure out what the difference in

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height will be well let's label each part of this diagram obviously the

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length of the string doesn't change so that makes our hypotenuse L which means [hypotenuse labelled L on triangle]

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that the vertical part of this triangle equals L minus H and that 30-degree

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angle will be our theta so for our angle the vertical side will be the adjacent

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side and we know the length of the hypotenuse and yeah, sohcahtoa we've got

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ourselves a cosine situation so here's the equation the cosine of theta equals

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L minus H the adjacent side over L the hypotenuse now we just have to solve for

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H we'll get everything on the same level by multiplying each side by L giving us

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L times cosine of theta equals L minus H now we can isolate H the easiest way to

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do that is to add H to both sides and then subtract L times cosine theta from

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each side and that leaves us with this equation H equals L meters minus the

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product of L times the cosine of theta now I can just plug in our numbers H

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equals 0.5 meters minus the product of 0.5 meters times the cosine of 30 [Formula appears]

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degrees when we use our calculator to find the cosine of 30 degrees there

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we've only got one sig fig to deal with here so well that becomes 0.07 meters

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also known as seven centimeters now that's bringing trig into the physical

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world so there you go some old guy with a weird name did all this math few [Right triangle discussing trigonometry and Pythagoras appears]

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thousand years ago and here we are doing it on some piece of technology that

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would blow his ancient Greek mind that's right Pythagoras you might have taught

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us that the shortest distance between two points is a straight line but we

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know better the shortest distance between two points is

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whatever our GPS tells us to do [Right triangle hands smart phone to Pythagoras]

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