Now lets add a circle through the point and with the origin as its center. Lets start with a point. P(x,y) r What does a point really mean well it means that it is x units left or right from the origin and y units up or down from the origin. Y ø X Therefore we can find the trig ratios of any point from the origin (reference angle) just by its x and y. Sin ø = y/r csc ø = r/y Cos ø = x/r sec ø = r/x tan ø = y/x cot ø = x/y In the unit circle the radius is 1 so sin = y csc = 1/y cos = x sec = 1/x tan = y/x cot = x/y

r y x x2 + y2 = r2 x and y represent the coordinates of the point on the unit circle and the radius is always 1 (unit circle)

Lets look at some types of problems that will use all the information about trig ratios and the unit circle. Finding all the trig functions exact values given a point on the unit circle: Example: ( ½ , -√3/2) Sin ø = y/r csc ø = r/y Cos ø = x/r sec ø = r/x tan ø = y/x cot ø = x/y Sin ø = -√3/2 Cos ø = ½ Now use the triangle ratios to find the rest: Tan ø = y/x or sin/cos -√3/2 ÷ ½ = -√3 csc ø = 1/sin = -2/√3 sec ø = 1/cos = 2 cot ø = 1/tan = -1/√3

Finding all the trig functions exact values given a point on the unit circle: Practice #15 ( -2/5 , -√21/5) Remember from Friday the x value is the value of cosine and the y value is the value of sine so… Sin ø = Cos ø = Now use the triangle ratios to find the rest: Tan ø = y/x or sin/cos csc ø = 1/sin = sec ø = 1/cos = cot ø = 1/tan =