What is a unit circle?

A circle centered at the origin that has a radius of one (x² + y² = 1).

 

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 60Ëš 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!!!