Special Angles in relations to Trigonometric Ratios

Hey gang!!! Super excited to finally get this over with so I'll make it short and simple. We are still pretty new in this unit so there is not a lot of things I can talk about and I'll try my best to not repeat anything that have been already said. As the title suggest I will be looking at special Angles, and Trigonometric Ratios.

When working on determining exacts values of a trigonometric ratio, we are often face with most angles on the unit circle that are based on reference angle of either 30º- 60º- 90º The "special" nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions.

30º-60º-90º Triangle Pattern Formulas

        Labeling:

H = hypotenuse

LL = long leg (across from 60º)

SL = short leg (across from 30º)

 

http://www.youtube.com/watch?v=7B1yrRLSRT8&noredirect=1

Find the exact value of 

           sin 45º + cot 45º

            

              

.

*An easy way of differentiating values for 30º-60º is in 30º adjacent is √3 and opposite is 1. While in 60º the values switches where adjacent if 1 and opposite is √3.*

For further help, if you already haven't done this, here is a small table of values of Trigonometric Functions in Isosceles Triangle, and Quadrantal Angles. 

 .

Having done this brings you closer to finding the exact value where the next step is applying the appropriate Trigonometric Functions. With this info you should be able to figure out how to determine the exact coordinates for each P(θ) given. 

 

Watching these videos really solidified my understanding of this unit because I get to hear how others explain it and I am able to piece together things much better having already learned the concept.

http://www.youtube.com/watch?v=TcHrKhFBzB0

https://www.khanacademy.org/math/trigonometry/basic-trigonometry/basic_trig_ratios/v/using-trig-functions

So that's pretty much it for what I have to say, I know its not much but these are things that made it easier for me to understand what is going on. I hope you guys can make something out of this and good luck in your studies!

The Unit Circle

Darryl here, I guess its my turn to blog. This short Blog will be about The Unit Circle. Most information are taken from the Booklet.

A unit circle is the circle with its center at the origin and a radius of 1 unit.

 The figure here shows all the measurements of the unit circle:

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

SINE, COSINE AND TANGENT

We can recall cosθ as the x-axis and sinθ as the y-axis

P(θ) is used to point the terminal point, where the terminal arm of angle θ intercepts the unit circle. P(θ) can be defined as P(x,y).

Booklet

sin θ = O/H => sin θ = y/1 = y

cos θ= A/H => cos θ = x/1 = x 

tan θ = O/A => tan θ = y/x  

In this case, P(x,y) = P(cosθ,sinθ)


Pythagoras Theorem says that In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The equation of a unit circle is x ² + y ² = r ², and since  x=cos and y= sin we can say that 
cos² θ+sin²θ = 1² ,Note* cos θ²+sin θ² = 1² is different from cos² θ+sin²θ = 1²

The Cast Rule

The cast rule is a really simple way of knowing which quadrant a trigonometric function will be located.

Cast Rule - Simple Writes.

Quadrant I is located at the top right corner (A) where all are positive.

Quadrant II is located at the top left corner (S) where only sine is positive leaving Tangent and Cosine negative, this means if you were to have a positive Sine value, it will be allowed in Quadrant I and Quadrant II.

Quadrant III is located at the bottom left corner (T) where T is positive.

Quadrant IV is located at the bottom right (C) where C is positive.




If you were to have a x-coordinate as ⅔, to figure out which Quadrant it is located on. We must find the y value = sin, P(⅔,?)

Using the unit circle equation = x ² + y ² = r ²

as the Radius is 1 unit.

x ² + y ² = 1

(⅔)^2  + y ² = 1
4/9 + y² = 1
y ²  = 1 - 4/9
y ² = 9/9 - 4/9

√y²= ± √5/√9 or ±√5/3
P(⅔,-√5/3) will lay on Quadrant 4 since its sine value is negative and cos being positive. 

                             

P(⅔,±√5/3)

That's basically all I have to say.

Although here is a method that I found on the internet.

Click Here

And a video.

That is all
Graph Cosine and Sine Functions

Graphing Parent Sine and Cosine Functions Precalculus Trig Functions

Degrees and Radians

Hey class Garrett here with a blog post about Degrees and Radians.

When measuring you first want to determine whether the rotation of the arm is clockwise or counterclockwise. If it is clockwise then the angle will be considered to be negative, if counterclockwise the angle will be considered positive.

How do we find the number of radians that are in a circle? Well lucky for you there is a simple formula that you can use to determine it.The formula is a= θr where a is the arc length r is the radius and  θ is the angle in radians. One thing to note is that this formula only works with radians not degrees so be sure not to use it in the wrong situation.

You may ask what is a radian? Well a radian is a unit of measurement just like a degree. When your arc length and your radius are equal that is one radian.

 Personally i find radians confusing and hard to understand but there are a few tricks you can use make them easier. One is the formula above another is the equation to convert radians to degrees. This formula is π/180 all you have to do is multiply the degrees with this formula and you get radians.

A sample calculation would be 60 degrees convert to radians

The first thing you do is write the equation so you get 60=π/180 then you just simply multiply.
So you should get π/3

Here is an simple radian circle to look back on if you need help

 

 

Lets do a harder example now -5π/3 converted to degrees

 

okay so the first thing you do is -5*180/3

 

This will be -300 degrees now you can make your graph and don't forget to go clockwise with your arrow because this is a negative degree question.

Some important radians to remember are:

π/4 which is 45 degrees
π/3 which is 60 degrees
π/2 which is 90 degrees
π    which is 180 degrees
2π  which is you guessed it 360 degrees

Thanks for taking the time to read my blog post and i hope this helped
everyone at least in a little way to better understand radians and degrees.