When Do I use Sin, Cos or Tan?
Summary
TLDRThis 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
📚 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.
🔍 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.
📉 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.
📐 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.
💌 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.
Mindmap
Keywords
💡Trigonometric functions
💡Right triangle
💡Hypotenuse
💡Acute angles
💡Trigonometric ratios
💡30-60-90 triangle
💡Sine
💡Cosine
💡Tangent
💡Memory device
💡Algebra
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
تصفح المزيد من مقاطع الفيديو ذات الصلة
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