Some help with trigonometry in general...

  • #1
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Main Question or Discussion Point

i was trying to learn some trigonometry from some basics ...

after understanding terms like ,

f(x) = sin x , and

y = sin x ...

i was also getting sort of comfortable with the notations of functions ...

i was also wondering , where to improve from here ... ?

any good books , sites ? tutorials ... ?
 

Answers and Replies

  • #2
Simon Bridge
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They will all just state the notation to use and then use it - if you are "uncomfortable" with notation for functions then they won't help.

The foundation trig is just based on angles in the unit circle.
A unit circle is one that has a radius of 1 unit.
The size of an angle is the distance around the circumference of a unit circle that is inside the angle.
The trig functions are just the names of distances using this definition ...
For instance: the tangent line to the unit circle is any line that touches the circle at only one point.
If you draw a line through that point and the center of the circle, you can make an angle to that line by drawing any other line through the center of the circle.
The tangent of that angle is the length of the tangent line inside that angle.

There are similar definitions for sine and secant.
Once you have that down - you can do trig.

Other than that, just google for "introduction to trigonometry".
 
  • #3
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this was all very confusing for a very long time ... you know , the opposite / hypotenuse ... adjacent / hypotenuse ... all those things ...

looked so dull and uninteresting to me , until i found this picture online ...

2f281bb.jpg


suddenly , the quadrants and the sine , cosine , opposite / hypotenuse ... adjacent / hypotenuse etc is a bit more interesting ...

4b2fcb2.jpg


i have few more important questions to ask ...
when does a function like ...


f(x) = sin x

becomes like a repetitive cycle , and becomes like a wave like motion ?

fccf344.png


for example ...


how to graph a sine function ?

http://www.dummies.com/how-to/content/how-to-graph-a-sine-function.html

Knowing how to graph trig functions allows you to measure the movement of objects that move back and forth or up and down in a regular interval, such as pendulums. Sine functions are perfect ways of expressing this type of movement, because their graphs are repetitive and they oscillate (like a wave)
f(x) = sin x

It repeats itself every 2-pi radians ...

This repetition occurs because 2-pi radians is one trip around the unit circle — called the period of the sine graph — and after that, you start to go around again. Usually, you're asked to draw the graph to show one period of the function, because in this period you capture all possible values for sine before it starts repeating over and over again. The graph of sine is called periodic because of this repeating pattern
i really need a better understanding of this part ...

is it really going to repeat itself for an infinite period ??
 
  • #4
Simon Bridge
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When I teach this, I get students to measure the sine and tangent lengths and plot the results on a graph.
You should probably do this.

Draw a unit circle: radius = 1.
Draw a line going through the center of the circle, passing through the sides ... right through: this is the "base line" for the angles.
Ignore your protractor: this is a distraction.

Draw a line starting from the center and passing through the side of the circle. This is the angle line.
The length around the circumference between the base line and the angle line is the size of the angle.
Notice there are two angles: these angles add up to half the circumference of the unit circle.
Call the smaller angle ##\theta##, then the bigger one is ##\pi - \theta## because the circumference of a circle radius 1 is ##2\pi##.
This is also why radians are the most natural way to do angles.

OK, add some labels
- the center of the circle is point O;
- the base line intersects the side of the circle at points A and A';
- the angle line intersects the circle at point B ... so ##\theta## is the angle AOB;

You can draw the tangent line to the circle at point A.
This intersects the angle line at point C.
The distance |AC| is called "the tangent of theta" or ##\tan\theta## for short.
You can draw the angle line in at different angles and make a plot of ##\tan\theta## vs ##\theta## and see how it behaves.
You can measure ##\tan\theta## with a ruler! For this exercise it is useful to make your unit circle quite large.
What happens as ##\theta## approaches ##\pi/2##? How about just past ##\pi/2##?

The distance |OC| is called "the secant of theta" or ##\sec\theta## for short.

You can draw a chord from point B, that makes a right-angle to the base line. It intersects the circle at B and B', and intersects the base-line at point D.
The distance |BD| is twice |BB'| ... and is called "the sine of theta" or ##\sin\theta## for short.
You can repeat the exercise above for the sine... use different angles and measure the sine.
 
  • #5
198
10
thanks a lot for a lot of explanations ...

there is lot in it to think about ...

in the meanwhile , let me see where i can make small progresses ...

this discussion was very helpful , in moving forward positively ...
 
  • #6
198
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i have few more questions about little bit more advanced trigonometric functions ...

i don't know if i should be asking about it here .. or if i should start a separate thread about it ?

my question was mostly about functions such as these ...

f(x) = sin (kx − ωt)

y = sin (kx − ωt)

is it simply a representation of a trigonometric function in a graph ? or does such functions has some sort of practical application , such as some sort of wave propagation ?


this was the way i was trying to understand it , from the usual basics... to graph like cyclic representation to some sort of wave propagation that goes from the theoretical representation of a function in a graph to physical wave propagation?



71bd973.jpg

a0a8cc9.jpg

cf75da3.png

until this part , it looks like some sort of wave like representation in a graph ...

which isn't actually going anywhere ...

527c57c.jpg



but then , what happens here ??

07b5420.png





i don't exactly understand the operations here ??
 
  • #7
ProfuselyQuarky
Gold Member
814
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this was all very confusing for a very long time ... you know , the opposite / hypotenuse ... adjacent / hypotenuse ... all those things ...

looked so dull and uninteresting to me , until i found this picture online ...

proxy.php?image=https%3A%2F%2Fi.imgsafe.org%2F2f281bb.jpg
May I inquire how that picture made trig more interesting...?
 
  • #8
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  • #9
ProfuselyQuarky
Gold Member
814
526
substitute.png
This made me laugh out loud. By far, it's the most useful application of math I've ever seen! :oldlaugh:

Nevermind @rosekidcute. Carry along. You're clearly just being prepared for the next time you come across three raptors :biggrin:
 
  • #10
22,097
3,279
This made me laugh out loud. By far, it's the most useful application of math I've ever seen! :oldlaugh:

Nevermind @rosekidcute. Carry along. You're clearly just being prepared for the next time you come across three raptors :biggrin:
It's very funny. But question 1 and 2 are very well-posed problem too. Especially 2 is fun to solve. :biggrin:
 
  • #11
ProfuselyQuarky
Gold Member
814
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It's very funny. But question 1 and 2 are very well-posed problem too. Especially 2 is fun to solve. :biggrin:
Sure they are, but I'm not going to bother solving them because my hand is literally numb from decomposing too many partial fractions.

**Anyway, sorry for derailing the thread. Please resume back to what you were doing.
 
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  • #12
198
10
ProfuselyQuarky ,

first of all i am not really a trigonometry expert ... i have lots of things to improve from my basics ... which is sometimes why i like to use pictures ... it clearly gives me a wider picture of the things i am trying to solve ...

for example ...
this is me trying to solve

f(x) = sin x

i am not sure , what are all the sort of information i need ... to solve such a question ... i am only trying to refresh a few things from my old syllabus ...

the information should be usually looking like this ...

2f0e69f.jpg


lots of opposite / hypotenuse , adjacent / hypotenuse , opposite / hypotenuse ...

sometimes its a bit hard to imagine where this is all going .... and looks all the same ...

then i came across this picture ...

6947183.jpg


and it was sort of fun trying to think something inside a triangle ...

and when you try to do this again , opposite / hypotenuse , adjacent / hypotenuse , opposite / hypotenuse ...
you should be able to see some difference ...

like one of these raptor jokes , micromass posted...

2e1e611.png


i guess the way you view it , looks a bit more interesting ... like if you start calculating the opposite / hypotenuse , adjacent / hypotenuse , opposite / hypotenuse ...
some parts of that human in the middle of those raptors ... gets eaten last ???

i am not sure about that at this point , or it could be the way i see things ... or i might be looking at it the wrong way ...
 
  • #13
ProfuselyQuarky
Gold Member
814
526
Okay, it's fantastic that you founds away to help you with trig. It takes creativity to associate something like that with the image, so if it helps you, then shoot :smile:
 
  • #14
198
10
thanks ,

i am happy that my trigonometry is improving somehow .. even though its like a very slow activity ...

i was also wondering if anyone could help me with few more basic questions from trigonometry ...

i am like self learning this from scratch ...



when you talk about a function like ...

f(x) = sin x

the x , after the sin is the representation of angle in theta ...


at the same time it also means , opposite / hypotenuse ...

so what does this question really mean ... ?

d3e6ecc.jpg


f(x) = sin x

(the x , after the sin is the representation of angle in theta) and it also means ( opposite side length/ hypotenuse side length )

how are these things related ? i mean the angle in theta , the opposite side length / hypotenuse side length ...
 
Last edited:

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