4.2, 4.4 – The Unit Circle, Trig Functions The unit circle is defined by the equation x 2 + y 2 = 1. It has its center at the origin and radius 1. (0, 1) (1, 0) 1 (0,  1) (  1, 0)

4.2, 4.4 – The Unit Circle, Trig Functions If the point (x, y) lies on the terminal side of θ, the six trig functions of θ can be defined as follows: (x, y) y θ x A reference triangle is made by “dropping” a perpendicular line segment to the x-axis. r 2 = x 2 + y 2 r (−, +) (−, −)(+, −)

4.2, 4.4 – The Unit Circle, Trig Functions Evaluate the six trig functions of an angle θ whose terminal side contains the point (−5, 2). (−5, 2) 2 −5

4.2, 4.4 – The Unit Circle, Trig Functions For a unit circle (radius 1) 1 (1, 0) 1  (x, y) sin  = y cos  = x tan  =

4.2, 4.4 – The Unit Circle, Trig Functions 1 (1, 0) 1

4.2, 4.4 – The Unit Circle, Trig Functions

Find the six trig functions of 0º (1, 0) r = 1

4.2, 4.4 – The Unit Circle, Trig Functions Deg.Rad.SinCosTan 0º0º º 45º1 60º 90º10undef. 180º  0−10 270º−10undef. 360º 22 010 Summary

4.2, 4.4 – The Unit Circle, Trig Functions Basic Trig Identities ReciprocalQuotientPythagorean sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ cot 2 θ + 1 = csc 2 θ Cofunction sinθ = cos(90  θ) tanθ = cot(90  θ) secθ = csc(90  θ) Even cos(  θ) = cos θ sec(  θ) = sec θ Odd sin(  θ) =  sin θ tan(  θ) =  tan θ cot(  θ) =  cot θ csc(  θ) =  csc θ

4.2, 4.4 – The Unit Circle, Trig Functions Use trig identities to evaluate the six trig functions of an angle θ if cos θ = and θ is a 4 th quadrant angle. sin 2 θ= 1 − cos 2 θ 4 5 −3

4.2, 4.4 – The Unit Circle, Trig Functions For any angle θ, the reference angle for θ, generally written θ', is always positive, always acute, and always made with the x-axis. θ θ'

4.2, 4.4 – The Unit Circle, Trig Functions For any angle θ, the reference angle for θ, generally written θ', is always positive, always acute, and always made with the x-axis. θ' θ

4.2, 4.4 – The Unit Circle, Trig Functions For any angle θ, the reference angle for θ, generally written θ', is always positive, always acute, and always made with the x-axis. θ θ'

4.2, 4.4 – The Unit Circle, Trig Functions Find the reference angles for α and β below. α = 217º β = 301º α' = 217º − 180º = 37º β' = 360º − 301º = 59º 37º 59º

4.2, 4.4 – The Unit Circle, Trig Functions The trig functions for any angle θ may differ from the trig functions of the reference angle θ' only in sign. θ = 135º θ' = 180º − 135º = 45º sin 135º=  sin 45º =  = cos 135º = − tan 135º = −1 θ θ'θ θ'

4.2, 4.4 – The Unit Circle, Trig Functions A function is periodic if f(x + np) = f(x) for every x in the domain of f, every integer n, and some positive number p (called the period). 0, 2π sine & cosine period = 2π secant & cosecant period = 2π tangent & cotangent period = π

4.2, 4.4 – The Unit Circle, Trig Functions sin = tan =

Find the exact value of each.