General Angle Ratios: Help & Explanation

[tumelo]

can someone please help with the General Angle ratios,i don't understand the division into 4 quadrants,I would like someone to expain in detail

 

[tiny-tim]

Welcome to PF!

Hi tumelo! Welcome to PF!

can someone please help with the General Angle ratios,i don't understand the division into 4 quadrants,I would like someone to expain in detail

(What do you mean by "General Angle ratios"? )

The division of angles into the 4 quadrants is just a matter of convenience.

As you know, quadrants 1 to 4 are 0º to 90º, 90º to 180º, 180º to 370º, and 370º to 0º.

So if x and y are both positive, then (x,y) (-x,y) (-x,-y) and (x,-y) lie in quadrants 1 to 4 respectively.

To put it another way, the quadrants correspond to cos and sin of the angle both being positive, one negative and one positive, both negative, and one positive and one negative.

There's nothing special about it.

 

[tumelo]

Okay that I understand,wht I really wanted to say was what are Sine,Cosine and Tan,what do they really mean,because if I take example Sine 180(deg)=0,wht did Sine do o make 180(deg) zero?

 

[symbolipoint]

You are ready to use results of an internet search for "unit circle". A ray of length 1 is rotated from the positive x-axis using endpoint of the ray at the origin. The sine is y/1, and the cosine is x/1.

 

[tiny-tim]

Okay that I understand,wht I really wanted to say was what are Sine,Cosine and Tan,what do they really mean,because if I take example Sine 180(deg)=0,wht did Sine do o make 180(deg) zero?

In a right-angled triangle:

cos = adjacent/hypontenuse = adj/hyp

sin = opposite/hypontenuse = opp/hyp

tan = opposite/adjacent = opp/adj​

(and adj and opp go negative if they go "the wrong way" … eg for the 2nd quadrant, adj, which is always along the x-axis, is going the "wrong way", so it's negative, so cos is negative, but opp, which is always parallel to the y-axis, is still positive )

These are what cos sin and tan really are! ​

 

[tumelo]

thnks for the reply.

lets have a look at this equatio

sin x=opp/hypo

the right side of the equation are numbers(opposite and hypotinuse) which is okay,now the problem is the left side,it means tht the function(if i may call it tht) sine multiplied by a given angle to give a number equal to the right hand side,so sine could be a constant i don't know,tht is wht i want to find out

its just like pi in area of a circle right,we know tht it has t be multiplied by the radius squared to give the area,and we also know tht its a constant whch is approximately 3.142. so wht about the sin,cos,and tan,wht are they?

 

[tiny-tim]

Hi tumelo!

… now the problem is the left side,it means tht the function(if i may call it tht) sine multiplied by a given angle to give a number equal to the right hand side,so sine could be a constant i don't know,tht is wht i want to find out

Yes, sine is a function, but no you don't "multiply" it …

sine is like √ (or log or e^…) …

√x isn't something-times-x (eg, √9 isn't something-times-9), and similarly sin(x) also isn't something-times-x …

it really is just a function, and there's no way of short-cutting it!

 

[sportsstar469]

Hi tumelo! Welcome to PF!


(What do you mean by "General Angle ratios"? )

The division of angles into the 4 quadrants is just a matter of convenience.

As you know, quadrants 1 to 4 are 0º to 90º, 90º to 180º, 180º to 370º, and 370º to 0º.

So if x and y are both positive, then (x,y) (-x,y) (-x,-y) and (x,-y) lie in quadrants 1 to 4 respectively.

To put it another way, the quadrants correspond to cos and sin of the angle both being positive, one negative and one positive, both negative, and one positive and one negative.

There's nothing special about it.

370? i thought a circle was 360?

 

[VeeEight]

Yes, I assume that was a typo

 

[tumelo]

wow! thanks Tiny-Tim,i think now i am getting somewhere

first let me say tht i was wrong by using the "multiply" word i just wanted to emphasise the fact tht sine does "SOMETHING" to the angle to give a NUMBER(not an angle) which will be equal to the right hand side if we take an example of sine x˚=opposite/hypotenuse

just like Tiny-Tim gave an example of √.we know tht the √ means the "root" of a number. so wht about sine,cosine or tan

 

[tiny-tim]

370? i thought a circle was 360?

he he …

i meant 270º ! ​

just like Tiny-Tim gave an example of √.we know tht the √ means the "root" of a number. so wht about sine,cosine or tan

Sorry, but cosine(x) isn't "part" of x, like √(x) is …

(though it is the real part of e^ix, if that makes you any happier)

most functions are just themselves, and there's nothing more to be said about it …

they have the Popeye raison d'être …

I am what I am, and that's all that i am!

 

[tumelo]

hey tiny-tim,i have an attached document tht u shud check it out and explain to me using sine for the angle α

 

[tiny-tim]

hey tiny-tim,i have an attached document …

attached to what? ​

 

[tumelo]

Tiny-tim

Ohhhh my bad sorry bout tht,anyway let's drop it i found some material tht will help me understand the origins of trigonometric funcions,thanks for the replies though,hope to communicate wit hu on another topic, i am doing a big project this side

 

[emyt]

a right angle can be completely determined by angles and sides.. if you have an angle of 45 degrees for example, a side directly "opposite" to that angle will be determined as well. think about making a triangle with an angle, and see if you can make the opposite side larger or smaller without changing that angle.

now you can see where "SOH CAH TOA" comes from..if you don't know, they are the ratios of the sides:

Sine: Opposite/Hypotenuse (triangle side opposite to the angle over the largest side) Cosine: Adjacent/Hypotenuse (triangle side "next" to the angle that isn't the largest side over the largest side) and Tangent: Opposite/Adjacent

note that an angle can be shared between smaller and larger triangles. this is why we consider the ratios, it turns out that there is a determined ratio between opposite, adjacent and hypotenuse sides whenever an angle is determined.
By convention, trigonometry generally refers to a "unit circle", where the radius - therefore the hypotenuse, is always 1. So if you have sin(x) = ? it's really just the length of the "opposite side", since the hypotenuse is 1. (anything divided by 1 is just itself)


anyway, a function is more than just things you can do with addition and multiplication.. et c. think of a function as a "rule" or a "map" that has a value stored for something you give. in this case, if you give the sine function a value, it gives back a ratio (that is determined by the angle you have given)


example:
consider a right-angle triangle with a 45 degree angle. since all 3 angles should add up to 180 degrees, you know that the other side is also 45 degrees (90+45+45). Also, keep in mind that the hypotenuse should equal to one..

okay, let's put what I just said to the test:

what is sin(45 degrees)? sin(45degrees) = squareroot(2)/2. and since you know that the other angle is 45 degrees as well, sine of that angle is also sqrt(2)/2. you can think about it like this if you are only considering the sine function. however, cosine(45 degrees of the same angle) is also defined to be sqrt(2)/2.

so from the Pythagorean theorem, we know that if we are right, we should get 1:

(sqrt(2)/2)^2 + (sqrt(2)/2)^2 = ? ---> 2/4+2/4 = 1

 

[tumelo]

thank u very much for tht infomation,hope to get more from u on other topics!