What are the basics of trigonometry? And why are we learning about this in a physics course? Both good questions. In this video, you'll learn about sines, cosines, tangents, and more... and about how these concepts are applied to physics. Sorry - math and physics go hand-in-hand. Let's just hope they regularly use Purell.

Courses | Physics |

Language | English Language |

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

vector quantities well a scalar quantity is it's something

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

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

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

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

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

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

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

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

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

world of trig sohcahtoa not quite the classic like Shazam

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

down all right what makes a triangle a

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

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

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

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

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

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

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

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

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

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

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

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

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

trigonometry's' greatest-hits is the cosine

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

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

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

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

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

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

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

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

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

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

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

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

Babylon thousands of years before Pythagoras but whatever

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

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

mc-squared the Pythagorean theorem says that the

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

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

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

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

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]

B squared Pythagoras you clever dog you thanks for

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

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

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

have corresponding inverse functions those would be the arcsine, arccosine

and arctangent functions you might also see them written like the

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tangent and tangent will cancel each other out

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

know better the shortest distance between two points is

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