Physics: The Basics of Trigonometry

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.

CoursesPhysics
LanguageEnglish Language

Transcript

01:03

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

01:08

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

01:12

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

01:17

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

01:21

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

01:26

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

01:30

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]

01:35

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

01:39

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

01:43

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

01:47

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

01:51

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

01:55

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

02:00

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

02:05

vector quantities well a scalar quantity is it's something

02:10

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

02:16

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

02:21

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

02:27

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

02:31

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

02:34

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

02:39

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

02:43

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

02:49

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

02:53

world of trig sohcahtoa not quite the classic like Shazam

02:58

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

03:03

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

03:07

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

03:11

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

03:16

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

03:21

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

03:26

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

03:31

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

03:36

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

03:40

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

03:44

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

03:49

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

03:53

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

03:58

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

04:02

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

04:06

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

04:10

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

04:16

down all right what makes a triangle a

04:19

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

04:23

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

04:28

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

04:33

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

04:38

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

04:41

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

04:45

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

04:49

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

04:54

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

04:58

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

05:03

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

05:06

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

05:12

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

05:16

trigonometry's' greatest-hits is the cosine

05:20

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

05:24

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

05:29

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

05:34

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

05:41

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

05:46

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

05:50

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

05:54

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

05:59

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

06:03

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

06:08

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

06:12

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

06:16

Babylon thousands of years before Pythagoras but whatever

06:19

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

06:22

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

06:27

mc-squared the Pythagorean theorem says that the

06:30

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

06:35

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

06:39

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

06:44

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

06:48

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]

06:52

B squared Pythagoras you clever dog you thanks for

06:56

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

07:00

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

07:05

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

07:09

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

07:12

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

07:16

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

07:21

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

07:25

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

07:29

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

07:33

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]

07:38

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

07:43

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

07:45

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

07:51

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

07:53

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

07:58

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

08:02

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

08:07

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

08:13

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

08:18

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

08:21

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

08:25

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

08:30

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

08:34

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

08:38

have corresponding inverse functions those would be the arcsine, arccosine

08:42

and arctangent functions you might also see them written like the

08:47

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

08:52

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

08:56

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

09:01

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

09:08

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

09:12

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

09:17

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

09:24

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

09:30

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

09:35

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

09:38

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

09:45

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

09:50

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

09:54

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

09:58

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

10:03

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

10:08

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

10:14

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

10:18

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

10:22

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

10:27

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

10:31

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

10:37

tangent and tangent will cancel each other out

10:39

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

10:44

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

10:48

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

10:52

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

10:56

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

11:00

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

11:04

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

11:09

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

11:13

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

11:17

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

11:21

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

11:25

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

11:30

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

11:32

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

11:36

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

11:40

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

11:45

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

11:50

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

11:54

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

12:00

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

12:06

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

12:11

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

12:17

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

12:21

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

12:26

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

12:31

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

12:34

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

12:38

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

12:43

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

12:48

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

12:52

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

12:55

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

12:59

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

13:03

know better the shortest distance between two points is

13:06

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