Unit 4 - Circular Functions - The Unit Circle I

Hi, I'm Navdeep!

I'm going to scribe about the second topic in Unit - 4 "Circular Functions" and the topic is called The Unit Circle I.

A unit circle is the circle with its centre at the origin and with a radius of one unit. Equation of the unit circle is x² + y² = 1.

Its standard position starts at (1,0).

Positive distance is measured in a counterclockwise direction; and negative distance is measured in a clockwise direction. 

Terminal Point is where the terminal arm of an angle ϴ intercepts the unit circle. For every arc. length ϴ on the unit circle, P(ϴ) is unique. 

 

Always remember, if r = 1, then:

sinϴ                      =            O/H            =              y/r                = y/1 = y

cosϴ                     =            A/H            =              x/r                = x/1 = x

tanϴ                      =            O/A            =              y/x               = sinϴ/cosϴ

So, that's why P(x,y) = P(cosϴ, sinϴ), and since x² + y² = 1, we have cos²ϴ + sin²ϴ = 1.

But if the circle has a centre with its origin and radius is other than one 1, the equation would be 

x² + y² = r².

CAST Rule

For example: Given that sinϴ = -⅘ and cosϴ = ⅗, in which quadrant does ϴ lie?

 We know that sin is negative and cos is positive given in the question. So now according to the CAST Rule ,we know that we have sin negative and cos positive in Quadrant IV. 

If we have to determine the coordinates of on the unit circle when we have one of the coordinate is given and have to find the other so we can use
x² + y² = 1.

For example: the y-coordinate is -⅟₂ and the point is in Quadrant III and determine the x-coordinate.

We can use this equation to solve for x by substituting the value of y:
x² + y² = 1x² + (-⅟₂)² = 1x² + ⅟₄ =1x² = 1 - ⅟₄x² = 3/4x = ±√3/4

Special Right Triangles

Now we have to understand and memorize the reciprocal trigonometric functions too with their new names as follows:  

Just don't try only to memorize them but try to understand that how all of them are related to each other!

Reference Triangles

There are two reference triangles: one with the angles of 45+45+90 degrees and the other one with 30+60+90 degrees. 

Let's look at the example of a reference triangle with the angles of 45+45+90 degrees:

Quadrantal Angles

There are three different quadrantal angles and can be written in degrees, in exact values and in approximate values. Take a look at the following example:

So in this example, first circle shows the angles in degrees, second shows in radians as exact values, and the third one shows in decimals as approximate values.

UNIT CIRCLE

       So, this is the unit circle and we have a similar diagram as above in the booklet too and if anyone has not finished filling in this unit circle due to any kind of confusion or then hope it is helpful. 

      Hope all above would be helpful and have a great day!!

The End!