4.2 - Trigonometric Functions:
The Unit Circle
Penguin says "Circles are nice!"
What is a unit circle?
A circle centered at the origin that has a radius of one (x2 + y2 = 1).
The unit circle is divided mainly into twelve equal arcs whose angles follow the radian/
degree measures of the special right triangles (degrees of 30 or/6, 60 or /3, 45 or /4, and
multiples of these up until 360 or 2). The degree/ radian values can be used to find the an (x,y)
coordinate that lies on the circle. Using the (x,y) coordinate, you can find the six trigonometric
functions.
Penguin says, "Yeah! !"
You can use the unit circles special right triangles to find the trigonometric functions:
These are 30-60-90 triangles. Because they are 30- 60 -90 special right triangles, the sides will always
be 1- 2- √3 ( where 1 is opposite the 30˚,/6, angle, 2 is the hypotenuse, and √3 is the remaining side).
For a 45-45-90 special right triangle on the unit circle, the sides will always be 1- 1- √2
( where √2 is the hypotenuse and the other sides are 1).
penguin says, "These triangles are super, just like ME!"
There are six trigonometric functions that follow the unit circle:
1) Sine (sin) Sinθ = opposite/ hypotenuse (y/radius)
2) Cos (cos) Cosθ = adjacent/ hypotenuse (x/radius)
3) Tan (tan) Tanθ = opposite / adjacent (y/x)
To remember these functions you can use the acronym Soh Cah Toa
Penguin says, "Soh Cah Toa? Is that a type of fish?"
These are the reciprocals of the first three functions respectively:
4) Cosecant (csc) Cscθ = hypotenuse/ opposite (radius/y)
5) Secant (sec) Secθ = hypotenuse/ adjacent (radius/x)
6) Cotan (cot) Cotθ = adjacent/ opposite (x/y)
Penguin says, "A reciprocal is something filped upside down."
*Remember that radius = 1 for the unit circle
Penguin says, "Surprise! It's time for..."
EXAMPLE PROBLEMS!!!!
Ex 1) Find the six trig functions for 60˚ using a special right triangle:
Step 1) draw a triangle and lable the degrees and sides
Step 2) find the trig functions using their definitions
Sinθ = (opp/hyp) = √3/2 Cscθ = (hyp/opp) = 2√3/ 3
Cosθ = (adj/hyp) = 1/2 Secθ = (hyp/adj) = 2
Tanθ = (opp/adj) = √3 Cotθ = (adj/opp) =√3/ 3
Ex 2) a: Find the six trig functions at the real number θ = /6
Step 1) Find the corresponding point (x,y) on the unit circle. By looking at the unit
circle at the top of the page, the corresponding point to θ = /6 is (√3/2, 1/2).
Step 2) Use the definition of the trig functions to find the values.
Sinθ = (y/radius) = 1/2 Cscθ = (radius/y) = 2
Cosθ = (x/radius) = √3/2 Secθ = (radius/x) = 2√3/2
Tanθ = (y/x) = 1/√3 Cotθ = (x/y) = √3
b: Find the six trig functions at θ =5/4
Step 1) Follow the same procedures as part "a." The (x,y) coordinate for θ =5/4 is (- √2/2, - √2/2)
Sinθ = (y/radius) = - √2/2 Cscθ = (radius/y) = - √2
Cosθ = (x/radius) = - √2/2 Secθ = (radius/x) = - √2
Tanθ = (y/x) = 1 Cotθ = (x/y) = 1
Ex 3) Evaluate the six trig functions for θ= -/3
Step 1) Moving clockwise around the unit circle (because of the negative sign), θ= -/3 cottesponds to the pont
(x,y) = (1/2, -√3/2)
Step 2) Find the six trig functions
Sinθ = (y/radius) = √3/2 Cscθ = (radius/y) = 2√3/ 3
Cosθ = (x/radius) = 1/2 Secθ = (radius/x) = 2
Tanθ = (y/x) = √3 Cotθ = (x/y) =√3/ 3
Happy Holidays!!!
Comments (1)
Anonymous said
at 8:35 pm on Jan 8, 2008
I love the color, and the penguins are cute. Great job. 20/20
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