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How to learn Trigonometry functions and formulas easily?

Conquer the applications of Trigonometry today!

Did you wish you had a Trigonometry cheat sheet to master this subject? Don’t worry! All your questions and queries will be answered by our subject specialist, Nitesh Negi, in a LIVE Trigonometry Webinar on 4th February 2021 at 8:30 am (PST).

Whether you love or hate Trigonometry, we have answers to all your questions. Here is your chance to find out answers to the sin, cos, tan problems that have plagued you since the beginning.

Who all will benefit from this webinar?

Come one, come all! If you are a working professional, teacher, parent homeschooling your child or anyone with an interest in Trigonometry, we’ll be glad to have you on board.

Moreover if you are a high school student this webinar is for you! Students from 9th-12th grade will be the biggest achievers as all your queries on identities and theorems will be answered by an expert professional.

How will this webinar help you?

If you wish to upgrade your Mathematical skills and learn trigonometric functions, identities, and applications, this webinar is for you. Moreover, get expert tips on how to tackle tricky questions.

You will also learn what role Trigonometry plays in our daily lives. Overall, by reviewing the fundamentals and learning the study hacks, you can surely score better grades in your exam. Get ready for an exciting session where you can master the basics of the subject in no time.

What can you expect?

Be prepared for a 30 minutes intensive session highlighting basic concepts and applications of trigonometry. Nitesh, your tutor will also conduct 10 minutes one-on-one questions and answers round. So attendees, rest assured you stand a chance to get an answer for all your questions.

What’s more! We will have MCQ (Multiple Choice Questions) round for the live audience. No need to hesitate, ask all your doubts and we will make sure you get your answers.

Trigonometry Webinar Transcript

Hello Everyone

Thank you for...Thank you so much for joining the webinar

for learning the fundamentals and applications of Trigonometry with me

I'm Nitesh Negi and I am your Math tutor for this webinar.

In these 45 minutes, we will spend the first 30 minutes on learning the concepts of Trigonometry

And in the end, we will give 10-15 minutes for the Question & Answer round and for your doubts also

And this webinar is presented by the TutorEye.

This TutorEye is an online tutoring platform that aims to provide result-oriented tutoring to students from K-12

And our goal is to periodically present webinars for students addressing the latest education trends impacting online tutoring and education industry

So, let's just start it with our topic

And our today's topic is Trigonometry.

As you know, Trigonometry is used in a variety of fields.

As you can see here, we can use this in by measuring the height of the bridges, or the tall buildings, even this flight - the slope of those lines- how much length it should be

We can find this by using Trigonometry.

So, let's get started with the introduction for the Trigonometry.

"Trigonometry" is derived from the 3 words which is "Tri", which means three

"Gon", which means sides, and "Metron" means measure.

So, Trigonometry means the measure of the 3 sides

And the relationship with the angles of the Triangle

We have to be sure that it should always be a triangle to be a right angles triangle, not every other triangle

So this is the main thing here.

Now, the uses of Trigonometry.

How we can use Trigonometry in our daily lives and what are the applications of it?

As we have already discussed, let us have some more applications

So, we have..as we have already discussed,

The height of the buildings we can find, even we can find the height of the mountains by using Trigonometry concepts.

We can use this in constructions, in the navigation industry

It is vitally used in Physics and Engineering

It will always be used in Marine Biology

In Marine Biology, we always use this as a...to find the length or the height of any mammal or fish in the...sea

And in Criminology it is also used.

And in Math itself, it can be used to simply find the complex equations and integrals of Mathematics.

So this is all about Trigonometry and all its applications.

Now, let's go on to the Trigonometry fundamentals.

Trigonometry Webinar by Nitesh Negi Transcript

So, this is the most important thing in the Trigonometry

The Trigonometry Ratios

So, in Trigonometric ratios, before we start with

let us learn some things which are very much useful in this.

As you can see here, we have a Right-angled Triangle ABC.

And it has an angle 90 degrees at B-vertex.

So, in all the questions, it will always be given that at which vertex the 90 degree angle will be formed.

And the other 2 vertices are: It's just two other angles.

And we will always say that to which angle we have to actually find the Trigonometric ratios.

From this Triangle, I have assumed that we will find the Trigonometric ratio for that angle which we will be formed at the vertex C, so that is why I have mentioned it as a Phi.

And now, we should know that as you can see here, opposite, adjacent, and hypotenuse.

Hypotenuse, as you know already is the longest side of a right-angled triangle, which will always be the opposite side of the 90-degree angle.

And, these opposite and adjacent things are interchangeable according to the vertex.

As you can see here this C-vertex. So, AB side is opposite to it, so that's why we have said to it as the opposite. And the BC side is adjacent.

But, if I want to find the opposite for the angle A, or the angle at A, so its opposite side will be BC for that. And adjacent will be AB.

So, now let us learn, what are the Trigonometric ratios.

So, we have 6 Trigonometric ratios.

The first one is Sin

The sin is basically the ratio of the opposite and the hypotenuse side of a triangle.

And in this particular case, it will be AB over AC

And we have a reciprocal sign for this sign, which is Cosecant.

So, Cosecant and Sin are reciprocal for each other.

For Cosecant B, AC over AB i.e. hypotenuse over opposite.

So, now the next will be Cosine.

Cosine is BC over AC, which is adjacent over hypotenuse.

So, you have to learn this.

And now the reciprocal will be Secant and this Secant is the hypotenuse over adjacent.

Which is AC over BC.

So, these are the 4 and we have 2 more

1 is Tangent. Tangent will be Opposite over Adjacent

And the Cotangent will be Adjacent over Opposite.

So, these are the Trigonometric ratios which you should learn.

Just let me know if you have any doubts so far, or does everything makes sense so as we can proceed further.

I know that most of the students, find difficulty in learning these ratios, so we have a short trick or a trick method for that

We will discuss that also.

So, just let me know if you have any (questions).

And for asking this, you can raise your hand in this...you have the "raise you hand" option in the "more" option of your Zoom meeting.

So, you can raise your hand for that.

So, if you are finding a difficulty to learn that what should be the sine ratio, what should be the cosine ratio, or tangent ratio, you can always learn this statement.

If you can learn this statement, you can easily learn what should be the ratio of this.

As you can see that the statement says that "Student Of Henry Can Ask Him To Oppose Anxiety"

So, we just have to take the first letter of every word.

So, the first 3 letters are S, O and H

So, S stands for the Sine, O stands for the Opposite and H stands for Hypotenuse.

So, it will easily be learnt that Sine is Opposite over Hypotenuse.

In the second line, you can see the letters, which are Bold is C, A, and H.

So, it will be Cosine is equal to Adjacent over Hypotenuse.

And Tangent. Tangent is Opposite over Adjacent.

So, like this we can easily learn what should be the Trigonometric Ratios.

And we just have to learn 3 because we know that the other 3 are the reciprocal of these.

So, in this way, we can find all the Trigonometric Ratios by learning this one statement.

So, I think this will help you a lot in learning this process.

So, is it clear everything to all of you?

If so I am guessing that everyone has got this because no one is asking anything.

So, let's jump to the next one.

So, let's solve this question which is regarding the Trigonometric ratios.

The question is that in the Triangle LMN, right-angled at M, so see that they have given that which vertex we should have the right angle. At M, we should have a right angle. And they have given the 2 sides that LM will be 11ft and MN will be 60ft. So, we have to find all the Trigonometric ratios for angle L.

So, here we are also given that for which angle you have to find the Trigonometric ratio. As we have discussed. So, first of all we have to draw a triangle LMN and we have to show that all these things that 90 degree angle will be at M and LM will be 11 ft and MN will be 60 ft. So these are the two..

But, here you can see we have LN, we do not have the LN side, so we have to find this.

And how we can find?

For this we can use the Pythagoras theorem.

The famous Pythagoras theorem, you should already know that.

If not, you can learn it here.

So, Pythagoras theorem basically is we always the square of the hypotenuse

In this case, the hypotenuse is LN will be equal to the sum of the square of the other 2 sides.

That means LM square and MN square.

If we square these sides and add them, and we only square the LN side

with those 2 will be equal.

So, according to Pythagoras theorem, LN square should equal to LM square + MN square

So, we have to just put in the values for LM and MN, so it is 11 square and 60 square.

And we know that 11square is 121 and this 60 square is 3600.

When we will be adding these, we will be getting 3721

But that's the value of LN square, but we have to find the value for only LN, it's not the square of that side, we have to find the value for LN only.

So, we have to take the square root of both sides.

So, if we're taking the square root, we'll be getting 61 and,

So the value for the 3rd side, which is the hypotenuse, in this case, will be 61 ft.

And now, we can easily find all the values for Trigonometric Ratios of angle L.

So, first of all we have to decide, which side should be opposite and which side should be adjacent.

So, angle L, so as you can see, this is angle L. So, MN will be opposite to it and LM will be adjacent.

And 61 is already, you know that its hypotenuse, because in this Pythagoras theorem we have used that.

So now, everything will be fine and very much easy for us to find that.

So, let's find the first ratio which is Sin ratio.

Sin will be opposite over hypotenuse and here it will be MN over LN.

That means the answer should be 60 over 61

This is the answer for the Sin

And if we can find the value for this Sin, we can easily find the value for the Cosine also

So, that will be adjacent over hypotenuse.

But, before that, we should find the value for the reciprocal for the Sin, which is the Cosecant.

So, the Cosecant will be simply 61/60.

And we have already discussed that Cosine will be adjacent over hypotenuse which is 11/60.

And the reciprocal of Cosine. Reciprocal of Cosine is Secant.

And that will be 61/11. So, that's an easy thing.

And now we have Tangent. So, the Tangent will be 60/11 because it's hypotenuse over adjacent, and Cotangent will be an inverse of this. That means 11/60.

So, this is the way to find the Trigonometric Ratios in any right-angled triangle if we have all the 3 sides.

But, we can have angle also.

If the angle is given then, how we can find the Trigonometric Ratios.

For that, we have to learn this table for standard angles.

So, these are the standard angles.

As this is 0 degree, 30 degree 45, 60 and 90.

For these, we must learn, the values for that, but other than this, if we have an angle like 43 or a 55 or a 75.

For that, we can use the calculator.

So, this is a calculator for... designed for specifically these kinds of things.

So here, we can use the concept and we can use this thing so in order to find the values.

As you can see, this thing. These 3 buttons, this is a Sin, Cosine, and Tan.

If you press Sin, you will get the function Sin. And you have to put in the angle and in this case, its 30, but you can put any value, just like say, 75 and you will get the answer for that.

So, its that easy.

So, this is the way through which we can find all the Trigonometric Ratios.

So, just let me know if all the things make sense to you. Everything about the ratios, if we can find using the sides, or using the calculator, or this table.

If you have any doubt with these things, you can ask me now.

Or, if you want to ask in the end, that can also be possible.

Just let me know if you have any doubt till now.

I'll be waiting for a few seconds for you.

I guess so...I guess you must be asking in the end all of you.

So, okay, let me...take a few examples.

Or look into the application of this thing.

So, now the applications as you can see..the questions will be used in real-life problems.

That is basically the application.

You can say that there is a question of the real-life problem and they have sent from the top of a building of height 545 meters. So, the height they have given for the tall building and the angle of depression.

So, here I want to make sure that you should know what is an angle of depression and the other thing we should learn is the angle of elevation.

So, before solving this question, I will be giving you some basic things.

Let us suppose that this is your eye-level parallel to the ground. Okay?

So, now suppose that you are here...the eye is here. I can make a small eye.

So this is our eye-level or the sight of our eye.

If we're seeing anything which is above this, that angle will be called as the angle of elevation.

If we're seeing downwards, so this angle will be the angle of depression.

So, we have to always make sure that we should measure those angles to the parallel eye-level to the ground.

So, these are the two things we should know about it.

To take now let us solve the question.

The question is the angle of depression from the small tower is 50 degrees.

So, we must draw the 45 degrees...uh 45 meters tall building and then we have a small tower of 15 meters, and the angle of depression is 50.

So, as you can see the dotted line will show the eye level which is parallel to the ground and the angle will be 50 degrees.

Now, they've asked that what should be the distance between the two buildings.

So, how we can do this?

We will use the concept of Trigonometric Ratios.

And how we can do that?

Let's suppose that distance is 'd'.

If the distance is d, so we can find simply that the Tangent of that 50-degree angle will be the opposite over adjacent.

And in this case, the opposite will be 30.

But some of the students will ask me that what should be the...um 30 meters is not given in this question.

So, you can observe that this 15-meter height of the building and 45 is the height of the tall building.

So, if we subtract 15 from that, we will get the height of that triangle which is formed for the 50-degree.

So that is 30. Okay?

So, that's how we will get the value for 30.

And, for 50-degree, it's not a standard question.

So that means, we must use the calculator in order to find that.

So, using the calculator, the Tangent value is 1.1917.

Now, it's very much easy for us to find the distance. We just have to solve this equation.

Solving this, we should divide 30 and the value of Tangent 50.

So, by dividing, we will be getting 25.1741.

And that's how we find the distance between the two buildings.

So, this is the process.

So we have to make a diagram for it, or a figure for it. So as we can solve the questions very easily.

So, if you do not frame the figure, we will not be able to imagine, how we should continue with the question.

So, let us take one more example.

So, now the next example is from marine biology.

So, John is a Marine Biologist and he wanted to measure the length of a blue whale.

So, here you can learn how we can measure the length of the blue whale as we know that they are the largest mammal in the world.

So, what we're doing...we are measuring the angle of depression of the head and tail using the clinometer.

Clinometer is the instrument by which we are actually finding the angles. Okay?

We do not find by other ways; we find by using the clinometer.

So, the angle of depression for the head and tail are 60 and 30 respectively knowing the depth at which the blue whale is swimming.

We know that at which depthy the blue whale is swimming, which is 85ft.

So, the angle of depression we'll know.

So, let's start how we can solve this.

So, first of all we have to frame a diagram for this.

So, the first thing we have to see is the position of the 'jaw'.

So, let us suppose because we must be in a ship or a cruise.

So, that means the angle should be measured from the ocean level or sea level.

So, that's the line of sea level and the depth that is 85ft at which this blue whale is swimming.

So, now the angle of depression of the head is 60degree.

So, I'll make that.

And this is the right-angled triangle forming.

So we can use Trigonometric concepts also.

And another thing we can use is this is also a 30-degree, which is the angle of depression for the tail.

And you might be wondering that 60-degree is written here.

So, for that, you must know that this line and this line are parallel to each other.

And the angle which I have already...have shown that.

This angle is 60-degree, so these 2 angles are alternate interior angles.

So, that's why I've taken this angle as 60-degree.

So now, we are just wanting this thing, we do not want anything else.

So, we have to find this BC.

If we can find this BC, we will be able to find the length of the blue whale.

So, let's get started with this.

So, we have 2 Triangles here.

One with an angle of 60-degree and one here with an angle of 30-degree.

So, we have to use the Trigonometric ratios for 60 and 30 itself.

Let us take in triangle AOB.

So, A must be this point in the triangle AOB.

We want to use that.

So, Tan 60-degree, should be that AO, that is the opposite for 60-degree and OB, which is the adjacent for that.

So, Tan 60-degree, we just putting the values for this.

So, Tan 60, we know from the table which we have shown you, its root 3.

And OA is 85.

We can easily find the value for the OB.

So, we have found the OB.

And in a similar way, exact similar way, we can find the value of OC.

You can see that AOC which we...which might be Tan 30 degrees which is 85 over OC

and Tan 30 degrees is basically one by root 3.

So, by this way, we are finding the OC = 100

And now in order to find the length BC, what we should do?

We should subtract OB from OC in order to find this length.

So, this is the way and if you subtract these two values, you will be getting the length of that blue whale.

So, that is come out to give 51.38413 and we can approximate this value up to 2 decimal places and that would be 51.38.

That's how we can use this concept in Marine Biology also.

So now, let us go to the Whiteboard and let's solve this question.

So, this is the question in which they have given that Henry wants to measure the height of a building.

So, in order to measure the height of a building, we must know some things.

So, first of all, we have to draw a diagram for it.

So, let's make a diagram.

So, here we have to find the height of a building, so that means, we must draw a line here and suppose that this line is representing the tall building.

He walks away 70 feet, so that means from this point he is walking away from this building. You can take either side, the left side or the right side.

I am taking the left.

And he's looking up and the angle of elevation is 43-degrees.

So, that means from here the angle of elevation should be 43-degrees.

So, let's just make this.

This is your 43-degree and this distance is 75 feet.

And, let us suppose that the height of the building is 'h'.

Now, we can use the same concepts which we have already learned that here we have the use the Trigonometric ratios in order to find this.

But now we have to decide by which Trigonometric ratio you have to use.

Now the 43h is an opposite and this 75 is the adjacent.

So, in order to solve this, we must use the Tan ratio or Cotangent.

Because those two ratios involve the opposite and adjacent.

So, let us use the Tan. I'm using the Tan.

So, Tan 43-degree is equal to h over 75.

And if you want me to name this first, so let us name this.

This is an A, B, and C.

So, Tan 43 will be AB over BC.

And the value for AB is h, and the value for this BC is 75.

Now, in order to find the value of h, we should first find the value of Tan 43-degree.

And which we can find using the calculator.

Using calculator, this Tan 43 will be approximately equal to 0.9325 and that will equal to h over 70.

And in order to find the h, we should multiply these two things which is 0.9325 and if we multiply these two things, we will get the answer which will be 69.9375 feet.

So, this is the height of your building.

And, that's how we can solve all the questions in Trigonometry.

And we just have to take care about the angle and the sides and we can solve all the questions.

So, just let me know if you have any doubt in this question or any doubt in previous questions which we have done.

Just let me know if you have any doubts.

I'm opening the doubts section for you.

Any doubt in any kind of topic.

Okay Ashutosh, you have a doubt.

You can unmute yourself and just ask your question

Well I'd just like to know that how do you calculate Tan 43 actually?

I don't get...

Tan 43...yeah...I'll be telling you that

Tan 43...43 is not a standard angle, so we cannot use that table in order to find that.

So, I've discussed that you can use the calculator for it.

So we have a scientific calculator and which we can...okay let me..

I can show you the slide for the..

Let me show the slide.

Yeah, so this was the slide which we can have.

In this slide we have discussed how we can find the Trigonometric Ratios.

So, let me revise that.

So, you can see that these are the three button in which one Sin is written, then Cosine and then Tan.

You just have to press Tan button and then your angle which is 43 degree and you will find the value for that.

Is it clear?

Yeah, that's right.

Thank you so much.

Ok. Thank you very much.

Any other question from any other participant?

So, I'm just recalling what we have done in this whole webinar.

We have learned the fundamentals, the ratios, Trigonometric Ratios, and how we can find the values according to the given files or according to given angle and then we have solved some examples or the application to the real life problems.

And now, once again we are very thankful for everyone's participation in this webinar.

We hope you found the content helpful and informative.

Please share your feedback by taking a short survey at the end of the webinar.

Your inputs will help us improve and present better content to you on an ongoing basis.

Also, all the attendees will get access to Trigonometric formulas along with some more MCQs we will get some more MCQs also to test your skills.

And thank you everyone and hope to see you again in the next webinar.

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