4.2, 4.4 – The Unit Circle, Trig Functions

1. 4.2, 4.4 – The Unit Circle, Trig Functions The unit circle is defined by the equation x2 + y2 = 1. It has its center at the origin and radius 1. (0 , 1) (1 , 0) 1 (0 , 1) (1 , 0)

2. 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. r2 = x2 + y2 (− , +) r (− , −) (+ , −)

3. 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. 4.2, 4.4 – The Unit Circle, Trig Functions For a unit circle (radius 1) 1 (1 , 0) 1 sin = y cos = x tan = (x , y) 

5. 4.2, 4.4 – The Unit Circle, Trig Functions 1 (1 , 0) 1

6. 4.2, 4.4 – The Unit Circle, Trig Functions

7. 4.2, 4.4 – The Unit Circle, Trig Functions Find the six trig functions of 0º (1 , 0) r = 1

8. 4.2, 4.4 – The Unit Circle, Trig Functions Summary

9. 4.2, 4.4 – The Unit Circle, Trig Functions Basic Trig Identities Reciprocal Quotient Pythagorean sin2θ + cos2θ = 1 tan2θ + 1 = sec2θ cot2θ + 1 = csc2θ Even cos(θ) = cosθ sec(θ) = secθ Odd sin(θ) = sinθ tan(θ) = tanθ cot(θ) = cotθ csc(θ) = cscθ Cofunction sinθ = cos(90  θ) tanθ = cot(90  θ) secθ = csc(90  θ)

10. 4.2, 4.4 – The Unit Circle, Trig Functions Use trig identities to evaluate the six trig functions of an angle θ if cosθ = and θ is a 4th quadrant angle. sin2θ = 1 − cos2θ 4 5 −3

11. 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'

12. 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

13. 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'

14. 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

15. 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 sin135º = sin45º =  = cos 135º = − tan 135º = −1 θ θ'

16. 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 sine & cosine period = 2π secant & cosecant period = 2π tangent & cotangent period = π , 2π

17. 4.2, 4.4 – The Unit Circle, Trig Functions sin = sin = sin = tan = tan = tan =

18. Find the exact value of each.