4.2 The Unit Circle! Whoa wait a sec, where does all that stuff come from?

Since our special right triangles (or commonly used triangles are 30-60-90 and 45-45-90, let's split our circle into increments of 30 (12ths of the circle) and 45 (eights of the circle).

Let's go on a unit circle adventure and find out!

Convert degrees to radians

fourths of full circle, halfs of half circle

think 90/360 = 1/4

eights of full circle, fourths of half circle

think 45/360 = 1/8

twelfths of full circle, sixths of half circle

think 30/360 = 1/12

Mind Blown

SOHCAHTOA

Review!

Special Right Triangles

What if the hypotenuse was 1?

45

45

45-45-90 Trianglel

Find sin(45)=

cos(45)= tan(45)=

Let's talk special right trianglesWith a hypotenuse of 1, find the missing side lengths using the Pyth. thm.

45

45

1

x

x

x 2

30

601

30-60-90 Triangle

Find sin(30)=

cos(30)= tan(30)=

Find sin(60)=

cos(60)= tan(60)=

Let's talk special right trianglesWith a hypotenuse of 1, find the missing side lengths using the Pythagorean thm.hint: think of an equilateral triangle

30

60

x2x

x 3

Now we can find the coordinates of each point on the circle

cos(30) = x/1x = cos(30)x =

sin(30) = y/1y = sin(30)y =

cos(45) = x/1x = cos(45)x =

sin(45) = y/1y = sin(45)y =

Remember the hypotenuse is 1!

(mind blown)2