1. These angles will have the same initial and terminal sides. x y 420º x y 240º Find a coterminal angle. Give at least 3 answers for each Date: 4.3 Trigonometry Extended: The Circular Functions 60º -120º Subtract 360º from 60º : - 300º Add 360º to 420º : 780º Subtract 360º from -120º : - 480º Add 360º to 240º : 600º

2. Coterminal Angles An angle of xº is coterminal with angles of xº + k · 360º where k is an integer. Assume the following angles are in standard position. Find a positive angle that is coterminal with: a. a 30º angleb. a -2π/3 angle a.For a 30º angle, add 360º to find the coterminal angle. 30º + 360º = 390º A 390º angle is coterminal with a 30º angle. b.For a -2π/3 angle, add 2π (same as 360º) to find the coterminal angle. - 2π/3 + 2π = 4π/3 A 4π/3 angle is coterminal with a -2π/3 angle.

3. Let P = (-3, -4) be a point on the terminal side of . Find each of the six trigonometric functions of . r x = -3y = -4 P =(-3, -4)  x y -5 5 5 Text Example The bottom row shows the reciprocals of the row above. -4 -3

4. Definitions of Trigonometric Functions of Any Angle If  is an angle in standard position, and let P = (x, y) be a point on the terminal side of . If r = x 2 + y 2 is the distance from (0, 0) to (x, y), the six trigonometric functions of  are defined by the following ratios. P = (x, y) r x y 

5. Paper Plate Unit Circle Evaluate, if possible, the cosine function and the sine function at the following four quadrantal angles and place them on your paper plate Unit Circle: (1,0) (0,1) (0,-1) (-1,0)

6. Definition of a Reference Angle Let  be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle  ´ prime formed by the terminal side or  and the x-axis. a b  a b P(a,b) Find the reference angle , for:  =315º Solution:  ´ =360 º - 315 º = 45 º 

7. More on the Unit Circle Paper Plate and Reference Angles Find a reference angle,, for each of the following angles (and place them on your Unit circle too): 30º 60º π/4 DAY 2

8. x y STUDENTS Quadrant II SINE positive (and cosecant) ALL Quadrant I ALL FUNCTIONS positive TAKE Quadrant III TANGENT positive (and cotangent) CALCULUS Quadrant IV COSINE positive (and secant) The Signs of the Trigonometric Functions If tan  0, name the quadrant that  lies. IV

9. Use reference angles to find the exact value of sin 135° Step 1 Find the reference angle 135 º terminates in quadrant II with a reference angle  ´ = 180 º – 135 º = 45 º x y 135° 45° Text Example The function value, sin 45 º, for the reference angle is sin 45 º = Step 2 Use the quadrant in which  lies to prefix the appropriate sign to the function value. Because the sine is positive in quadrant II, put a + sign before the function value of the reference angle. sin 135  = +sin45  =

10. 60º 30º Use reference angles to find the exact value of the following trigonometric function (sketch it): sin 5π/3 in QIV, Sine is negative in QIV -

11. Use reference angles to find the exact value of the following trigonometric functions (sketch it). a. cos 2π/3b. tan 225 º in QII, Cosine is negative in QII in QIII, Tangent is positive in QIII c. sec (-30 º ) in QIV, Secant is positive in QIV

12. Unit Circle

13. Find cos θ and tan θ by using the given information to construct a reference triangle: sin θ = 3/7 and tan θ < 0 θ is in QII 3 7

14. Trigonometry Extended: The Circular Function