Help with the unit circle please.

In summary, the conversation discusses confusion over using the unit circle to find the values of trigonometric functions and the relationship between different angles on the unit circle. It is explained that the y-coordinate represents sine and the x-coordinate represents cosine, and the values can be found by plugging in the angle measure in degrees or radians into a calculator. The conversation also briefly touches on using trigonometric expansion formulas to simplify expressions involving trigonometric functions.
  • #1
dejan
29
0
Hi there.
I'm really confused with this question, it's about the unit circle.
Firstly I was asked to look up the values of (sorry I don't know how to put in the symbol for PI-3.14) 'cos(PI/3)' and 'sin(PI/3)' ok I got them.
Then I'm told to use the unit circle diagrams (diagrams?) to show the connection between:
cos(2PI/3), sin(2PI/3) and cos(PI/3), sin(PI/3)
(Then again for cos(4PI/3), sin(4PI/3) and cos(PI/3), sin(PI/3)

What is confusing me is the 'sin' and 'cos'! I don't understand where they fit into the unit circle?? I can find 2PI/3 on the unit circle, but how do I know if it is for sin or cos??

And by connection, I firstly thought, 'Ok, PI/3 and 2PI/3 are basically both 60degrees from the positive y axis' but then that isn't right because I haven't considered cos or sin!

If someone could point me in the right direction, it would be a huge relief!
 
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  • #2
Ok, I don't know if there is some light shining through my mind...but I checked on the calculator some things.
I put the mode on 'degree' and found out that cos60 (60degrees being that PI/3) is 0.5 and sin60=0.866.
Now the 2PI/3 is 120degrees. I put cos120=-0.5 and sin120=0.866
And for the 4PI/3-240degrees. Cos240=-0.5 and sin240=-0.866

Of course the pattern or similarities are obvious...cos60/120 are the same, just in different quadrants...sin60/120 are exactly the same, sin60/240 are the same, just in different quadrants, and same for cos240.

I'm still confused...and I feel like something is trying to get through to me...I'm searching around, but still I can't figure this out?? The question says to use unit circle to show the relationship...so does that mean I need to draw a unit circle and draw the angles for this all on it?

I don't know:\
 
  • #3
The unit circle is such that the y-co-ordinate of an angle is the sine of that angle and the x-co-ordinate is the cosine of the angle. Thus, when you draw an angle, the mark the point where it intersects the unit circle. The y-co-ord is the sin and the x-co-ord is the cos.

This is because for any point, y = r*sin(theta) and x = r*cos(theta). For an point on the unit circle, r = 1.
 
  • #4
forgive me if what i will be explaining you already know. After reading you message i believe you don't understand what sin and cos are and somewhat of the unit circle.

A unit circle is a circle drawn on a Cartesian Plane. Such circle has a radius of 1. Your angle measures have been given in radians, if you feel more comfortable in degrees just as a quick reference you can change those measures by multiplying the radians x (180/pi).

Anyways, imagine that you wrap a "number line" around the circle. Well, they give you an angle measure and you want to know where the end of line (of the angle) lies on the plane. Points on the Cartesian Plane are given as (x,y). Well, cosine would be the x, and sine would be the y. Let's get some easy ones if they ask you for example what is the cosine of pi/2, well (pi/2) x (180/pi)=90 degrees... as i said the unit circle is a circle with a radius of one. So, that means that the endpoints of that line will be (0,1) since the question is asking for cosine(which is x) then cos= 0.

Now, let me help you with that 2pi/3...I will convert it to degrees if it helps you, however, you should learn the basic measures in radians of the unit circle. ok, (2pi/3) x (180/pi)= 120 degrees, which will be on the second quadrant. Well, if you have a Ti calculator, you should change the mode(if you are going to be pluging in angle measures in degrees, have it on degrees. If you are going to plug in measures in radians put it in radians) now if simply you type cos(2pi/3) or cos(120) you should get the cosine, meaning the x of that line of the angle. cos of 2pi/3 is -.5, and the sine is .866 or square root of 3/2... the cosine (x) is negative because the point lies on the 2nd quadrant.

Hopefully i was able to help you out and hopefully you will understand the unit circle.

p.s you should learn the sine and cos for the basic radians angle measures which are: 0, pi/6,pi/4,pi/3,pi/2,2pi/3,3pi/4,5pi/6,pi,7pi/6,5pi/4,4pi/3,3pi/2,5pi/3,7pi/4,11pi/6
 
  • #5
Thank you both of you! Some things were made clear.

So, I already have that -.5 and .866 etc...do I plot them on the unit circle and join the points?? Then I should see the connection between them?? I think all of them in the end will have the same angle??
 
  • #6
Ok I think I get that now!

I have another question, where I need to use trig formula's.
It says to expand this term and simplify:
cos(A)+cos(A+2PI/3) + cos(A+4PI/3)
I'm not sure where to stay with this one? Should I put together 'cos(A+2PI/3) + cos(A+4PI/3)'? And then simplify the 4PI/3 to get 2PI/3 so that i can get 1 for that part and then it should equal cos(A)??
 
  • #7
I think you have to use the trigonometric expansion formulas for cos(a+b) ... cos(a+b) = cosa.cosb - sina.sinb
 
  • #8
Yeah that's it! I figured it out:) Thanks!
 

1) What is the unit circle?

The unit circle is a circle with a radius of 1 unit that is centered at the origin (0,0) on a coordinate plane. It is used in trigonometry to represent the values of sine, cosine, and tangent for any given angle.

2) How do I use the unit circle to solve trigonometric equations?

The unit circle can be used to find the values of sine, cosine, and tangent for any angle. By memorizing the coordinates of key points on the unit circle (0°, 30°, 45°, 60°, 90°), you can easily determine the values of these trigonometric functions for any angle.

3) Is it necessary to memorize the unit circle?

While it is not necessary to memorize the unit circle, it can be helpful in solving trigonometric equations quickly and efficiently. However, you can always use a reference sheet or a calculator to find the values of sine, cosine, and tangent for any angle.

4) What are the key points on the unit circle and their corresponding angles?

The key points on the unit circle are (0,1) at 0°, (√2/2, √2/2) at 45°, (1,0) at 90°, (√2/2, -√2/2) at 135°, (0,-1) at 180°, (-√2/2, -√2/2) at 225°, (-1,0) at 270°, and (-√2/2, √2/2) at 315°. These angles are also known as the special angles in trigonometry.

5) Can you provide an example of using the unit circle to solve a trigonometric equation?

Sure, let's say we have the equation cos(x) = 1/2. By looking at the unit circle, we know that the angle corresponding to this value of cosine is 60° or π/3 radians. Therefore, x = 60° or x = π/3 radians.

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