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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Hi,

I have recently begun to learn trigonometry. I learnt the trigonometry table yesterday. However, I have a few major issues with it.

1. Why does the table begin at 0°? I know that θ is a angle of a triangle. And I know from my previous chapters that no triangle can have a 0° angle. Then how can θ be 0 in any case?

2. Why is there a 90° angle? Since trigonometry is about right-angled triangles, then this would mean the other angle is 0° (by sum of interior angles property), which, as I have stated before, I have learnt to be impossible.

3. Why are there columns for 180°, 270° and 360°? This would suggest the other angles are negative! Is this even a triangle anymore?

So to summarise, I have a problem with practically every element of the trigonometry table, excluding 30°, 45° and 60°.

It does not seem to agree with any of my previous knowledge at all!

Learning is fun - Exceptio probat regulam!

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 37,965

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,467

hi CurlyBracket

Most maths courses start with the right angled triangle definition for trig ratios. But later you get to learn that these definitions can be extended to allow the angle to be any value.

To do this a fresh definition is needed that is consistent with the 'old' triangle definition, but allows angles to be anything.

The page https://www.mathsisfun.com/algebra/trig … rants.html shows how this is achieved.

Once the 0 < angle < 90 restriction is removed it opens the door to lots of useful new areas of maths.

I did a quick 'google' and found this video that shows one application https://www.youtube.com/watch?v=HteDupD6rL4

and the BBC site https://www.bbc.co.uk/bitesize/guides/z … revision/1 shows another.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Hi Bob and ganesh,

Thanks for all the useful links. I tried my best to understand the 'Sine Cos Tan in 4 Quadrants ' part, but I find the fact that they skipped Coordinate Geometry at school is largely preventing me from doing so. So now my plan is to study some of that, and then return to Trigonometry again. I haven't watched the video yet, but I will as soon as I get some time off.

Thanks for your help! I'll reply to this thread if I face any more issues.

Learning is fun - Exceptio probat regulam!

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**Mathegocart****Member**- Registered: 2012-04-29
- Posts: 2,224

CurlyBracket wrote:

Hi Bob and ganesh,

Thanks for all the useful links. I tried my best to understand the 'Sine Cos Tan in 4 Quadrants ' part, but I find the fact that they skipped Coordinate Geometry at school is largely preventing me from doing so. So now my plan is to study some of that, and then return to Trigonometry again. I haven't watched the video yet, but I will as soon as I get some time off.

Thanks for your help! I'll reply to this thread if I face any more issues.

Hey CurlyBracket,

Best of luck on your mathematical journey! There might be twists and turns on the way, but it will be genuinely rewarding.

The integral of hope is reality.

May bobbym have a wonderful time in the pearly gates of heaven.

He will be sorely missed.

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Thanks, Mathegocart.

Yes, I'm sure it will be rewarding.

Learning is fun - Exceptio probat regulam!

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**Hicies87****Member**- From: Portsmouth
- Registered: 2022-01-22
- Posts: 12

I tried to learn, but the attempt ended in failure. Something I didn't get to study. After that, all the difficulties associated with trigonometry, I found using https://plainmath.net/secondary/geometry/trigonometry for this. I tried again to figure it out, but finally realized that it was not mine. I'd rather trust someone who understands something about this.

*Last edited by Hicies87 (2022-06-28 01:30:12)*

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,467

Would you both like a lesson, (from scratch), about trig functions? I'll be happy to try and put one together for you if you ask.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Hi Bob,

That would be very kind of you.

I find most sources, like textbooks, to either ramble about really basic knowledge, OR to skip many parts in a bid to be faster.

Learning is fun - Exceptio probat regulam!

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,467

hi CurlyBracket

Ok, I'll try a new approach which I hope will be intelligible and not too hard to follow. I'll do one step at a time; please reply after each step so I know whether to move on or spend more time on that step. Ask any questions you like about it.

Firstly, forget everything you've learnt about sines and cosines. I'll start afresh.

(1) Two dimensional graphs.

The mathematician Descartes invented this idea. Draw a horizontal line. This he called the x axis.

In the middle of the line mark a point, called the origin.

Draw a vertical line crossing the x axis at the origin. This is called the y axis.

To get to a point on the paper you can specify an amount to go across from the origin, and an amount to go up from there to reach the point. The x and y distances are called the coordinates of the point.

This image shows such a graph. The origin has coordinates (0,0). A is roughly the point (0.7,0) I actually chose A pretty much at random, so it's exact coordinates are not important here. To get to A you have to start at the origin, go 0.7 across and nothing up.

P is the point (0.7,0.7) Start at the origin and go 0.7 across to get to A and then 0.7 up to get to P.

Note: the angle PAO is 90 degrees because I went horizontally then vertically.

Now imagine that P is free to rotate around O, but always on that circle. The circle has radius 1, so the distance OP will always be 1 too.

But, as P rotates the distances OA and AP will change. When P is on the x axis it's coordinates are (1,0).

As P moves anticlockwise around O, OA will get smaller and AP will get bigger. In my diagram I've shown P when it has rotated about 45 degrees, and OA = 0.7; AP = 0.7

Imagine P carrying on rotating around O. When P reaches the y axis, OA = 0; AP = 1.

If P turns even further the x coordinate goes negative and the y coordinate starts to decline.

I lack the ability to make a moving image of this, but one already exists thanks to MathsIsFun. Have a look at the animation here:

https://www.mathsisfun.com/algebra/trig … raphs.html

I love just watching this. It shows P rotating around O and the triangle OAP changing shape as it does so.

Here's some questions to test yourself:

(a) What is the largest that AP can ever be?

(b) What is the most negative it can be?

(c) In my image the angle POA is 45 degrees and we know the coordinates. Just based on that and using symmetry for what other angles would you be able to give the distance AP? You should be able to find three possible answers here, one between 90 and 180, and two more where the angle is over 180.

(d) If P goes al the way round and starts a second circle we could record the angle by using numbers over 360. For example, P will be back to my image position when the angle of turn is 360 + 45 = 405. What is the next angle when P is in this position?

End of part (1).

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Hi Bob,

A big THANK YOU!!

Okay, I'll answer the questions now.

(a) What is the largest that AP can ever be?

(b) What is the most negative it can be?

(c) In my image the angle POA is 45 degrees and we know the coordinates. Just based on that and using symmetry for what other angles would you be able to give the distance AP? You should be able to find three possible answers here, one between 90 and 180, and two more where the angle is over 180.

(d) If P goes al the way round and starts a second circle we could record the angle by using numbers over 360. For example, P will be back to my image position when the angle of turn is 360 + 45 = 405. What is the next angle when P is in this position?

(a) 1 unit.

(b) -1 unit.

(c) I'm not about this one. But I know the value for AP at 0, 90, 180 and 270.

(d) 405 + 360 = 765. Oh wait, that sort of explains the Trigonometry Table thing! I see now.

Learning is fun - Exceptio probat regulam!

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,467

a, b and d all correct, well done.

Part c. If you reflect the line OP (when the angle is 45) in the y axis the height AP is the same. Measured from the +x axis, what is the rotation angle now?

Now reflect in the x axis for two more answers.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Ok, got it.

Then the known values for AP will be at 135, 225 and 315.

Learning is fun - Exceptio probat regulam!

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,467

Correct. So would you like lesson 2?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**CurlyBracket****Member**- From: Your Maths Textbook
- Registered: 2022-01-03
- Posts: 122

Yes, please!

Learning is fun - Exceptio probat regulam!

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,467

OK. So if you specify an angle, the point P rotates anticlockwise around the origin until POA reaches that angle and then AP is a measurable amount between -1 and + 1.

That makes this a function : input a number as the angle and get AP as the output. All angles are possible so the domain is (- ∞, + ∞) and the range is [-1,+1]

We can give this function a name. It is called the SINE function, often abbreviated to sin.

It makes an interesting graph and that has some useful applications. I tried 'googling' it and got this : "In real life, sine functions can be used in space flight and polar coordinates, music, ballistic trajectories, and GPS and cell phones.

But it can also be used to determine measurements is triangles.

Let's say you have a triangle OAP where OP = 1, and you want to know AP. You could look on the graph and get a value. But you'd need to know the angle POA as well. Using a graph is OK but it's only as accurate as the quality of the graph. But there are two other ways you can get a value for sin(angle). Before the days of everyone having access to a calculator, you could use a table to look it up. I still have my table book from my own school days. If I want sin(45) I can turn to the sine page, look down the angle column and read off the value as 0.7071

The table book only gave 4 figures. I know now that the true value for sin(45) has lots more figures than that, but, for most purposes 4 digits is enough. The table will allow one decimal place, so I could also get sin(45.8) for example, and by using interpolation, sin(45.85) ... interpolation means estimating a value between those given. In this case I would look for 45.8 and 45.9 and take the value half way between them.

When the first calculators were available they could only do +, -, x and ÷ so one still had to use the table book, but later they devised calculators that could make use of 'power series'. At this stage you don't have to know what that means but it resulted in calculators that could do sines, powers, and lots more. These were called 'scientific calculators' and nowadays most calcs are this type.

But what is OP isn't 1 ?

Suppose OP = 7. The triangle OAP can be enlarged by a scale factor of x7, so OP = 7, and AP is 7 times as long. If the angle is still 45 then AP now is 7 x 0.7071 = 4.9497

Using scale factors you can calculate AP for any size triangle, provided angle PAO = 90. There are ways to deal with triangles that lack a right angle; more on that later if you want.

So we have a formula for doing this: **AP = OP times sin(angle)**

And this can be re-arranged to give **sin(angle) = AP/OP** and **OP = AP/sin(angle)**

There's a word for this sort of calculation, trigonometry. It comes from the Greek for measurements in triangles.

Here's some calculations to try:

(1) In a triangle PQR angle Q is 90. PR = 5 and angle R is 60. Calculate PQ.

(2) In a triangle RST, angle S is 90, and angle R is 32. Calculate TR if TS = 10

(3) In a triangle ABC, angle B is 90.If AB is 5 and AC is 7, calculate the size of angle C. Now work out angle A.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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