When Do I use Sin, Cos or Tan?

Summary

This educational video script explains the basics of trigonometry, focusing on when to use sine, cosine, and tangent functions with right triangles. It clarifies that knowing two sides or angles allows calculating the rest. The script introduces notation for right angles and unknown angles (theta), and explains the hypotenuse, opposite, and adjacent sides. It uses a 30-60-90 triangle example to illustrate sine, cosine, and tangent ratios. The presenter also shares a memory device, 'Sohcahtoa,' to remember the functions and encourages viewers to practice with provided links.

Takeaways

• 📐 **Trigonometry Functions**: The video explains when to use sine, cosine, and tangent in the context of right triangles.
• 🔗 **Right Triangle Basics**: Knowing two elements of a right triangle allows you to calculate the rest, including angles, area, perimeter, and side lengths.
• 📚 **Trigonometric Functions Defined**: Sine (sin), cosine (cos), and tangent (tan) are defined in relation to the sides of a right triangle and an angle (theta).
• 📏 **Hypotenuse and Angles**: The hypotenuse is the longest side and is opposite the right angle. The two acute angles in a right triangle always sum up to 90 degrees.
• 📐 **Standard 30-60-90 Triangle**: The video uses a 30-60-90 triangle to illustrate the ratios of sides in relation to the angles.
• 🔢 **Sine Calculation**: Sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
• 📏 **Cosine and Adjacent Side**: Cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• 🔄 **Tangent Function**: Tangent of an angle is the ratio of the length of the opposite side to the adjacent side.
• 🔍 **Using Charts**: Trigonometric values for specific angles can be found on charts or calculated using a calculator.
• 📝 **Memory Aid**: The video suggests a memory aid ('sohcahtoa') to remember the functions: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), tangent (opposite/adjacent).

Q & A

What is the main focus of the short video?

The main focus of the video is to teach when to use cosine, sine, or tangent in the context of right triangles, rather than how to use them.

What can you calculate if you know two things about a right triangle?

If you know two things about a right triangle, you can calculate all the others, including angles, area, perimeter, and the length of the other sides.

What is the significance of the 'D' notation in a right triangle?

The 'D' notation signifies a right angle, which is 90 degrees or 1/4 of a circle.

What does the Greek letter theta (θ) represent in a right triangle?

Theta represents an unknown angle in a right triangle.

What are the three types of sides in a right triangle?

The three types of sides in a right triangle are the hypotenuse (the longest side opposite the right angle), and the legs (the other two sides, sometimes referred to as the opposite and adjacent sides).

What is the hypotenuse in a right triangle?

The hypotenuse is the longest side of a right triangle, which is always opposite the right angle.

What is the sine function in the context of a right triangle?

The sine function is defined as the ratio of the length of the side opposite the angle (theta) to the length of the hypotenuse.

How is the cosine function different from the sine function?

The cosine function is the ratio of the length of the adjacent side to the length of the hypotenuse, whereas the sine function is the ratio of the opposite side to the hypotenuse.

What does the tangent function represent?

The tangent function represents the ratio of the length of the opposite side to the length of the adjacent side for a given angle in a right triangle.

What is the memory device 'sohcahtoa' used for?

The memory device 'sohcahtoa' is used to remember the ratios of the trigonometric functions: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent).

How can you verify the correctness of a calculated hypotenuse length?

You can verify the correctness of a calculated hypotenuse length by checking if it is longer than the other sides of the triangle, as the hypotenuse is always the longest side in a right triangle.

Outlines

00:00

📚 Introduction to Trigonometry Functions

The paragraph introduces the fundamental concepts of trigonometry, focusing on the right triangle as the basis for understanding when to use cosine, sine, or tangent. It emphasizes that knowing any two elements of a right triangle allows you to calculate the rest, whether it be angles, side lengths, area, or perimeter. The trigonometric functions are tools for these calculations. The paragraph also explains the importance of recognizing a right angle in a triangle and introduces the Greek letter theta (θ) to denote an unknown angle. It outlines the relationship between the sides of a right triangle: the hypotenuse (opposite the right angle), the opposite side (across from the angle in question), and the adjacent side (next to the angle in question). The concept of a 30-60-90 triangle is introduced, showing how the ratios of the sides relate to the angles.

05:01

🔍 Understanding Sine, Cosine, and Tangent

This section delves deeper into the definitions of sine, cosine, and tangent in the context of a right triangle. It explains that the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. The paragraph uses the example of a 30-60-90 triangle to illustrate these concepts, showing how the ratios are consistent regardless of the actual lengths of the sides. It also touches on the idea that these trigonometric ratios are pure numbers, as units of measurement cancel out in the ratios.

10:04

📉 Trigonometric Functions and Their Applications

The paragraph discusses how to apply the concepts of sine, cosine, and tangent when you know the angle and the length of one side of a right triangle, but not the other sides. It uses the example of a 50-degree angle to show how you can find the hypotenuse using the sine function and how you can find the adjacent side using the cosine function. The paragraph also introduces the idea of using a chart or a calculator to find the values of these trigonometric functions for different angles. It emphasizes the importance of checking whether the calculated answers make sense in the context of the problem, such as ensuring the hypotenuse is the longest side.

15:04

📐 Practical Examples and Memory Aids

This section provides practical examples of how to use trigonometric functions to solve for missing sides or angles in a right triangle. It introduces a memory aid to help remember the relationships between sine, cosine, and tangent, using the phrase 'soh cah toa' to represent 'sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent'. The paragraph encourages the viewer to pause the video and consider which trigonometric function to use in various scenarios, such as when you know an angle and the length of one side and need to find another side or the angle itself. It also humorously suggests not to trust toads from Soho, which is a playful way to remember the order of the functions.

20:06

💌 Conclusion and Contact Information

The final paragraph wraps up the video by summarizing the key points about when to use sine, cosine, and tangent in trigonometry. It provides a light-hearted reminder of the memory aid introduced earlier and invites viewers to reach out to the presenter with questions or comments via email. This paragraph serves as a conclusion to the video, reinforcing the learning objectives and offering further assistance.

Keywords

💡Trigonometric functions

Trigonometric functions, specifically sine, cosine, and tangent, are the focus of the video. These functions relate the angles of a right triangle to the lengths of its sides. The video explains that knowing any two pieces of information (angles or sides) allows one to calculate the rest, which is central to the theme of understanding trigonometric relationships.

💡Right triangle

A right triangle is a triangle that has one 90-degree angle. The script emphasizes that knowing two sides (or one side and one angle) of a right triangle allows you to determine all other sides and angles. This is foundational for understanding trigonometry as it sets the stage for applying trigonometric functions.

💡Hypotenuse

The hypotenuse is the longest side of a right triangle, opposite the right angle. The video script uses the hypotenuse to explain how trigonometric functions work, such as sine (opposite/hypotenuse) and cosine (adjacent/hypotenuse), making it a key term in understanding the relationships between the sides of a triangle.

💡Acute angles

Acute angles are angles less than 90 degrees. The script mentions that all right triangles have two acute angles that sum up to 90 degrees. Understanding these angles is crucial for applying trigonometric functions correctly within the context of right triangles.

💡Trigonometric ratios

Trigonometric ratios are the relationships between the angles of a right triangle and the lengths of its sides. The video script explains sine, cosine, and tangent as ratios (e.g., sine is opposite/hypotenuse). These ratios are essential for solving problems involving right triangles.

💡30-60-90 triangle

A 30-60-90 triangle is a special type of right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The script uses this triangle to illustrate the fixed ratios of the sides, such as the sine of 30 degrees being 1/2, which is a specific example of applying trigonometric functions.

💡Sine

Sine is a trigonometric function defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The script explains how to use sine when you know an angle and the length of the opposite side but need to find the hypotenuse.

💡Cosine

Cosine is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. The video uses cosine to explain how to find the hypotenuse when you know the length of the adjacent side and the angle.

💡Tangent

Tangent is the ratio of the length of the opposite side to the length of the adjacent side. The script mentions tangent in the context of finding the length of the adjacent side when you know the opposite side and the angle.

💡Memory device

A memory device is a technique to help remember information. The script introduces 'Soh cah toa' as a mnemonic for remembering the ratios of sine, cosine, and tangent, which is a practical tool for recalling these key trigonometric concepts.

💡Algebra

Algebra is used in the video to solve for unknowns in right triangles using trigonometric functions. The script demonstrates algebraic manipulation to find unknown side lengths or angles when other information is provided, which is a practical application of trigonometry in problem-solving.

Highlights

Understanding when to use cosine, sine, or tangent in trigonometry.

Trigonometry allows you to calculate unknowns in a right triangle when you know two sides or angles.

The significance of the 'D' notation in right triangles, indicating a right angle of 90 degrees.

The use of the Greek letter theta (θ) to denote an unknown angle.

Explanation of the terms 'hypotenuse', 'opposite', and 'adjacent' in relation to a right triangle.

The role of the hypotenuse as the longest side opposite the right angle.

The concept that all triangles sum up to 180 degrees in their interior angles.

The standard 30-60-90 triangle and its properties.

The sine function defined as the ratio of the opposite side to the hypotenuse.

The cosine function defined as the ratio of the adjacent side to the hypotenuse.

The tangent function defined as the ratio of the opposite side to the adjacent side.

How to determine which trigonometric function to use based on the known values in a triangle.

The importance of checking if the calculated hypotenuse is indeed the longest side.

Using trigonometric functions to find missing angles when you know the lengths of two sides.

The mnemonic 'sohcahtoa' to remember the ratios of sine, cosine, and tangent.

A personal mnemonic using 'Soho', 'cahoots', and 'toads' to remember trigonometric functions.

Practical examples of applying trigonometric functions to solve for missing sides or angles in a right triangle.

Transcripts

this short video is about how to know

when to use cosine sine or tangent and

not necessarily an exercise in how to

use them but knowing when to use which

one if you are having some trouble with

such and you want some practice problems

dealing with sine cosine and tangent

I've pasted a couple of links for you

here on YouTube so you can go to those

and see how that works for you a super

neat thing about right triangles is that

if you know two things about a right

triangle then you can calculate all the

others if you know the lengths of two of

the sides called legs or hypotenuse if

you know the length of any two you can

calculate angles you can calculate area

the perimeter the length of the other

side so it just takes two and that's

what the trigonometric functions are

about so one of the things that's kind

of important here is that a right

triangle has this little D notation

that's right through here that signifies

this is a right angle this is 90 degrees

this is 1/4 of a circle 90 degrees and

if you see this little guy over here

this simply means that you're dealing

with an unknown angle or we're talking

about this angle right here and we use

this symbol right here theta to denote

an unknown angle so if we don't know

what angle we're talking about and we

don't know how much it is we'll just

call it theta and theta can be the other

acute angle C all right triangles have

90 degrees and two acute angles and

these two acute angles always measure 90

degrees so this Plus this is always 90

degrees and this is 90 degrees and all

triangles in the known universe have a

hundred and eighty degrees inside them

so if you add the interior angles of

these to come up with 180 degrees it's

the same way with a right triangle there

are certain names and words that are

used for right triangles and that is if

we're talking about this angle right

here this angle theta right here this is

the hypotenuse the hypotenuse is always

the longer

line in a right triangle and this is the

opposite side from the angle theta and

so if we're talking about this angle

this is the opposite side this is the

adjacent side adjacent means next to and

of course opposite means opposite and so

if we are talking about the different

angles that we have here and it doesn't

matter if you rotate it around the

hypotenuse stays the same hypotenuse is

always on the opposite side from the

right angle so here we have the the

right angle and the hypotenuse is the

longest line in any right triangle is

opposite from there doesn't matter how

we rotate it it still stays the right

angle and it still stays the hypotenuse

on the opposite side if theta is on the

other end of the triangle in other words

we were looking at theta here while we

go but now we have the line over here so

that means that we're talking about this

theta then this becomes the opposite

sign just because it's the opposite part

from theta this becomes the side or the

leg of the hypotenuse that is adjacent

to theta and the hypotenuse remains the

hypotenuse the hypotenuse never becomes

it never switches so the hypotenuse is

always the hypotenuse and again it's

always opposite the right angle so okay

so there a bunch of funny names to learn

but you can do it the name theta is just

a Greek letter that is used to denote an

unknown angle it looks like this the

word hypotenuse is just the name of the

longest line in a right triangle the

words opposite and adjacent aren't too

weird and the edges of a right triangle

that are not the hypotenuse are called

legs when the legs are different links

they're called the long leg and the

short leg I'll leave it to you to figure

out which one is the long leg okay we're

going to take a look here at a standard

30-60-90 triangle and I'll use it to

illustrate a couple of things for you so

if we're looking at a standard 30-60-90

triangle and we know that this angle

right here is 30 degrees this is the

opposite side and this is the hypotenuse

the ratio of

one unit here in two units here stays

the same for all 30 degrees so if this

is 30 degrees this length right here is

half is exactly half of the length of

the hypotenuse so if this is 5 units

long then the hypotenuse would be 10

units long the sine of 30 degrees is a

pure number the sine of 30 degrees it's

a fraction the sine of 30 degrees again

it's just a pure number the sine of 30

degrees is some length divided by some

length the unit's cancel if you have one

foot divided by 2 feet the feet are in

the numerator and the feet are in the

denominator and they cancel so it just

becomes a pure number for example sine

of 30 is equal to the length of the

opposite side divided by the length of

the hypotenuse sine of 30 is equal to

one unit divided by two units is 1/2 it

is that way for every 30-60-90 triangle

in the known universe so here we have

the 30 60 90 if the length of the

opposite side from the 30 degrees is 0.7

units long they can be feet they can be

miles it can be inches they can be yards

they can be millimeters doesn't matter

whatever this unit is and this is the

same unit over here it will always be

twice if we're talking about a 30-degree

angle so if I chop this off a little bit

this angle right here remains to be 30

degrees but what happens is the

relationship the ratio of the length of

this side to the length of the

hypotenuse the new hypotenuse is 2 to 1

so whatever the hypotenuse is it's twice

the length of this over here let's chop

it off a little bit more so if I went to

50 units over here the length of our

hypotenuse would be 100 units for all

30-degree triangles 30 degree right

triangles

and chop it off a little bit more and I

wind up with this is still 30 degrees

here in my sketch it's not a perfect 30

degrees I just sort of sketched it out

there and so we have five units and then

I button this would be ten units but

what happens if you don't know the angle

but you know the length of the two sides

any two sides let's take a look so here

we have we don't know the theta but we

do know the length of the opposite side

and we do know the length of the

hypotenuse that would be the sine of

theta is sine of theta is the opposite

divided by the hypotenuse so this is

just some length divided by some length

which becomes a pure number because the

units cancel out so the sine of theta is

the opposite divided by the hypotenuse

here we have the same theta we have the

same hypotenuse but we want to know

what's the length of the adjacent side

the length of the adjacent side is the

cosine of theta which is just some pure

number let's talk about this for just a

second right here so I have the cosine

of theta which is just some number it's

cleverly stored in your calculator or in

charts we'll get to that in a few

moments and it's just some number this

sign right here most people look at this

sign right here oh that means find the

answer no it doesn't mean find the

answer it means the amount of stuff on

this side is exactly equal to the amount

of stuff on that side so this cosine of

theta is just a pure number and this

right here the length of the adjacent

side divided by the length of the

hypotenuse is just a pure number because

if it's five feet over four feet the

feet are both in the numerator and the

denominator and they cancel so a pure

number and a pure number so let's take a

look at what happens if you know that

it's you know the angle theta or you

don't skew it you don't know the angle

theta and you know the length of the

opposite side and you know the length of

the adjacent side that would be what we

call a tangent function the tangent of

theta is just a pure number and if we

know the length of the opposite side and

we

oh the length of the adjacent side

they're just pure that's just a pure

number because it would be like four

feet divided by five feet and the feet

are in the numerator and they cancel so

let's take a look at the next slide that

I have here in force so all right

what happens if theta changes well if

theta jumps to the other side the other

acute angle there's are only two acute

angles in a right triangle it jumps to

the other side then the opposite side

now becomes what had been the adjacent

side and that's because this is opposite

the angle the hypotenuse it never

changes and so the sine the pure number

of sine is equal to the pure number of

the opposite divided by the hypotenuse

so now what happens if it's we've moved

the theta so now we have the adjacent

side here it's the side that's next to

theta its side that's next to the angle

and the hypotenuse so the cosine of

theta again a pure number is just the

adjacent side divided by hypotenuse

again a pure number and it's the same

way with the tangent so if I take the

tangent function here and I'm looking at

this the tangent function is just the

opposite divided by the adjacent so this

is my opposite side from my theta and

this is my adjacent side so this is just

some pure number cleverly hidden in your

calculator or on some chart it's just a

pure number like 1 or 2 or 0.7 and if we

take the length of the opposite side and

the length of the adjacent side and we

divide those out because this number

this slash right here adventure them

right here that just means division so

this is the numerator that's the

denominator this is just a fraction and

it's just a pure number so let's take a

look at the chart here so all right we

took a look while ago at the 30 degrees

and so here's here's our chart I've kind

of truncated it a little bit here but so

if we're talking about degrees it's the

first column here and this is the first

column and if we're looking for the sine

of 30 degrees well the sine of 30

degrees is exactly zero point five zero

zero zero correct a couple of decimal

places there it's cosine happens to be

this and it's tangent happens to be that

these are just pure numbers so we were

looking at the sine

of 10 degrees here's the 10 degree mark

and so this chart tells us that the sine

of 10 degrees is zero point one seven

three six that's just a pure number so

if we divided the opposite side by the

hypotenuse then we would come up with a

sine number of this and we could look at

the chart backwards and tell us that oh

that's ten degrees and we'll get into

that a little later okay so what happens

if you know the angle and the length of

one side well here we know the angle

this is 50 degrees and we know that the

length of the other side is one unit but

we don't know what the hypotenuse is and

we wanted to know what the hypotenuse is

so which function would this be it would

be the sine the sine of 50 degrees is

equal to the opposite divided by the

hypotenuse in this case 1/2 question

mark we don't know what it is now this

just becomes simple algebra and so we do

the algebra on that and we say okay so

we looked up on the chart the sine of 50

degrees go back to this see if we can

find that not again so all right so the

sine of 50 degrees I have it later the

sine of 50 degrees is the length 1 unit

divided by some other length and we

don't know what that is so we do the

little crunching on that I looked on the

chart and I have the sine of 50 degrees

happens to be zero point seven six six

zero correct to four decimal places we

could go further in the decimal places

usually sine cosine and tangent

functions aren't pure numbers aren't

even whole numbers or something like

that they usually are decimals they go

on forever and ever

so correcta four decimal places we're

looking here that we see that zero point

seven six six zero equals one the length

of the opposite side divided by the

hypotenuse now we simply then just come

in over here and we wind up doing some

algebra and we come up with that zero

point seven six six zero is exactly

equal to correct four decimal places

length of the hypotenuse directa four

decimal places is one point three zero

five four they have to ask ourselves

does that make sense

that's a habit you should get into with

all math problems does my answer make

any sense well we know that the

hypotenuse is the longest line in any

trying right triangle and so if this leg

over here happens to be one unit long

and we calculated the hypotenuse is

going to be one point three well that

kind of makes them sense that it's

longer if we got a shorter answer over

here we might need to go back and check

our math one little mistake and a whole

series of things can mean that your

hands forgets way off so always kind of

go back and check does my answer make

any sense take a look at another one so

what happens if you know the angle and

the length of one side well here we have

we're dealing with the 50 degrees excuse

me we're dealing with 50 degrees here we

know the length of the other side but we

don't know the length of this side right

here well it happens to be the tangent

function and we'll get into a little bit

later how it is that's the tangent

function and how you know that that's

the tangent function but the tangent

function is the opposite divided by the

adjacent side and so we were to crunch

on that a little bit let's say the

tangent is the opposite divided by the

unknown and so that the tangent of 50

degrees happens to be one point one nine

one eight correct four decimal places we

know the length of the opposite side but

we don't know the length of the adjacent

side so we crunch the numbers on that

and we come up with the tangent of 50

degrees happens to be one point one nine

one eight we know the length of the

other side and then we take a look at

this and we say okay does my answer make

any sense

well yes it does except for the drawing

seems to be a little off this is a

sketch don't ever trust the sketch this

is just a quick sketch that I made and

this isn't exactly fifty degrees here so

if this is one unit long this length

right here should

be shorter in my drawing but it's not I

intentionally made it a little bit

longer so that we could call attention

to the fact that sometimes you're

dealing with the sketch and you need to

trust the numbers not the sketch all

right so how did I get the tangent of

50s I looked on the chart and so here's

the degree mark right here and the

tangent correct to four decimal places

is one point one nine one eight that's

how I came up with that it's all so

cleverly stored in your calculator

calculators are different so you're

going to have to if you're if you're

using your calculator for this you have

to use your calculator and play with it

a little bit to see how that works so

how do I remember the trig functions so

I came up with my own little memory

device to help me remember the three

most common trig functions that is the

sine of theta is the opposite divided by

the hypotenuse and the cosine is the

adjacent divided by hypotenuse and the

tangent is the opposite divided by the

adjacent that's a whole lot of stuff to

try to remember and so I thought well

hmm all right well let's take a look at

the first letters of this s Oh H and see

a H and T Oh a and I wondered I asked

myself do I know any words that start

with Soh and CAH and Toa and I came up

with this so ho it's a word that I

already knew it's a place in London and

a place in New York City and it's just a

place Soho and then I already knew the

word cahoots see a H so cahoots so if I

remembered the word cahoots that would

help me remember cosine is the adjacent

divided by the hypotenuse and what word

did I know that started with a Toa well

that would be to'd Toa D and so I just

thought what do I know about Soho

cahoots and toads and the when you're

trying to remember something if you'll

just think it's something that's really

stupid it'll help you remember that

because it's easy to remember stupid

stuff so I thought of toads who

were in Soho who were in cahoots you see

goats means that cheap people and so

these two Toads right here are working

together to cheat this toad right here

and so here we have I mentioned that we

had some toads sitting in Soho uh in

these two in cahoots cheating the other

one so that's the way that I remember it

I just think of this piece of artwork

right here of Soho Liam toads in cahoots

to cheat each other but there are other

ways of thinking about it and probably

the most traditional way is the nonsense

word that was made up sohcahtoa sign is

the opposite divided by the hypotenuse

cosine is the adjacent divided by the

hypotenuse and a tangent is the opposite

divided by the adjacent all right well

by now you should have a good grasp on

the basics of windy use sine cosine or

tangent to solve missing links or angles

or right angles so we'll rip through a

few examples I'll show a problem and you

have to think to yourself which one do I

use if you need more time click anywhere

on the screen to pause the video then

click anywhere on the screen to start

the video okay what function would you

use I put a little hint for you right

here my words Soho cahoots and Toad so

if you wanted to find the missing length

right here and you knew this degree

measure what function would you use well

turns out you would use the tangent

function

you know the degree and the opposite you

don't know and the adjacent you do know

that's the tangent is the opposite

divided by the adjacent code what

function would you use

you would use the cosine function the

cosine or cahoots s I have written over

here cosine is the adjacent divided by

the hypotenuse and we don't know the

hypotenuse what function would you use

you would use the sine function here you

know the length of the opposite side but

you don't know the hypotenuse and you

want to know the hypotenuse what

function

you would use the sine function

good use the tangent function

you would use the cosine function

you would use the tangent function what

function would you use

you would use the tangent function what

function you would use the cosine

function what happens if you know both

links but not the angle what function do

you use to find the angle you would use

the tangent function what function do

you use to find the angle you would use

the cosine function what function do you

use to find the angle

you would use the sine function just

remember that you can't trust toads from

Soho they are always in cahoots to cheat

you or you can simply remember the

made-up nonsense word sohcahtoa feel

free to write to me at alan morris at

yahoo.com