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  1. Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions? Do Now: A circle has a radius of 3 cm. Find the length of an arc cut off by a central angle of 2700.

  2. Q II Quadrant I terminal side 90 <  < 180 0 <  < 90 terminal side initial side  t.s. Q III Q IV t.s 180 <  < 270 270 <  < 360 • An angle on the coordinate plane is in standard position when its vertex is at the origin and its initial side coincides with the nonnegative ray of the x-axis. Angles in Standard Position y x • An angle formed by a counterclockwise rotation • has a positive measure. • Angles whose terminal side lies on one of the axes is • a quadrantal angle. i.e. 900, 1800, 2700, 3600, 4500 etc.

  3. Q II Quadrant I 90 <  < 180 0 <  < 90  initial side - t.s. Q III Q IV 180 <  < 270 270 <  < 360 Co-terminal and Negative Angles y 3000 = x 600 • An angle formed by a clockwise rotation has a • negative measure • Angles in standard position having the same • terminal side are co-terminal angles.

  4. Q II Quadrant I 90 <  < 180 0 <  < 90 Q III Q IV 180 <  < 270 270 <  < 360 • Angles whose terminal side rotates more than one • revolution form angles with measures greater • than 3600. Angles Greater than 3600 y 4850 1250 x • To find angles co-terminal with an another angle • add or subtract 3600. 1250 and 4850 are co-terminal

  5. Model Problems • Find the measure of an angle between 00 and 3600 co-terminal with • 3850 b) 5750 c) -4050 • In which quadrant or on which axis, does the terminal side of each angle lie? • a) 1500b) 5400 c) -600 215o 315o 25o x-axis QIV QII

  6. hypotenuse side opp. cos  side adj. Unit Circle y 1 radius = 1 center at (0,0) cos , sin  (x,y)  x -1 1 -1

  7. Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions? Do Now: Find the measure of an angle between 00 and 3600 co-terminal with an angle whose measure is -1250.

  8. 3 Hypotenuse = 2  shorter leg Longer leg =  shorter leg Value of Sine & Cosine: Quadrant I y 1 radius = 1 center at (0,0) cos 600, sin 600 (x,y) 600 x -1 1 What is the value of coordinates (x,y)? 300-600-900 triangle Sine and Cosine values for angles in Quadrant I are positive.

  9. (x,y) 1200 side adj. 3 Hypotenuse = 2  shorter leg Longer leg =  shorter leg Sine values for angles in Quadrant II are positive. Value of Sine & Cosine: Quadrant II y 1 cos 1200, sin 1200 What is the value of coordinates (x,y)? 1 60º is the reference angle (180º-120º) 600 x -1 1 directed distance A reference angle for any angle in standard position is an acute angle formed by the terminal side of the given angle and the x-axis. What is the cosine/sine of a 1200 angle? 300-600-900 triangle Cosine values for angles in Quadrant II are negative.

  10. side opp. side adj. Value of Sine & Cosine: Quadrant III y 1 What is the value of coordinates (x,y)? What is the cosine/sine of a 2400 angle? 2400 directed distance 60º is the reference angle (240º-180º) x -1 600 1 directed dist. 1 (x,y) cos 2400, sin 2400 Sine and Cosine values for angles in Quadrant III are both negative.

  11. side opp. (x,y) Sine values for angles in Quadrant IV are negative. Value of Sine & Cosine: Quadrant IV y 1 What is the value of coordinates (x,y)? What is the cosine/sine of a 3000 angle? 60º is the reference angle (360º-300º) 3000 x -1 600 1 directed dist. 1 cos 3000, sin 3000 Cosine values for angles in Quadrant IV are positive.

  12. Unit Circle – 12 Equal Arcs

  13. Periodic Unit Circle – 8 Equal Arcs Negative Angles Identities

  14. y Quadrant II Quadrant I x Quadrant III Quadrant IV Value of Sine & Cosine in Coordinate Plane cos  is + sin  is + cos  is – sin  is + cos  is + sin  is – cos  is – sin  is – for any angle in standard position is an acute angle formed by the terminal side of the given angle and the x-axis. The reference angle:

  15. Model Problems • Fill in the table • Quad. Ref.  sin cos  • 2360 • 870 • -1600 • -36 • 13320 • -3960

  16. y 1 x -1 1 -1 Regents Prep On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 240 and find the exact value of sin 240o.

  17. Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions? Do Now: Use the unit circle to find: a. sin 1800 () b. cos 1800

  18. (-1,0) Model Problems Use the unit circle to find: a. sin 1800 () b. cos 1800 (x, y) = (-1, 0) sin 1800 = y = 0 cos 1800 = x = -1 180º - quadrantal angle

  19. sin  sin  cos  cos  = 1 -1 ( 1, tan) ( , )? Tan  radius = 1 center at (0,0) cos , sin  y 1 (x,y) tan 1  x -1 1

  20. Trigonometric Values + – + –

  21. Quadrant I Q II 90 <  < 180 0 <  < 90 Q III Q IV 180 <  < 270 270 <  < 360 Trigonometric Values - A C T S y S Sine is + A All are + x T Tangent is + C Cosine is +

  22. 1 y r y  x -1 x 1 -1 Reciprocal Functions Negative Angles Identities csc  = 1/y sec  = 1/x cot  = x/y denominators  0 Need to Knows When r = 1 sin  = y cos  = x tan  = y/x

  23. y (x,y) 1 1 sin  = y 2 2 450 x -1 1 cos  = x In a 450-450-900 triangle, the length of the hypotenuse is times the length of a leg. -1 (x) length of hypo. = Model Problems Using the unit circle, find cos 450 (/4) sin 450 tan 450 450-450-900 triangle A 450-450-900 triangle is an isosceles right triangle. therefore x = y cos  = sin 

  24. y 1 1 sin  = y 45º x -1 1 cos  = x = = -1 Model Problems Using the unit circle, find cos 45º(/4) sin 45º tan 45º (x,y) cos 45º = x tan 45 = 1 sin 45º = y

  25. 0º 0 30º /6 45º /4 60º /3 90º /2 sin  0 1 cos  1 0 tan  0 1 UND. Trigonometric Values for Special Angles Why is tan 90º undefined? What is the slope of a line perpendicular to the x-axis? = slope

  26. What is the cos 510º (17/6)? • cos 30º = • cos 510º= Model Problems What is the tan 135º (3/4)? • 135º is in the 2nd quadrant • 45º is reference angle (180 – 135 = 45) • tan 45º = 1 • tangent is negative in 2nd quadrant • tan 135º= -1 • 510º is in the 2nd quadrant • (510 – 360 = 150) • 30º is reference angle (180 – 150 = 30) • cosine is negative in 2nd quadrant ≈ -.866…

  27. Model Problems

  28. Model Problems Given: sin 68o = 0.9272 cos 68o = 0.3746 Find cot 112o • -0.3746 B) -2.4751 • C) -0.404 D) 1.0785 reference angle for 112o is 68o; 112o is in QII; tan and cot are negative in QII WHAT ELSE DO WE KNOW?

  29. Model Problems Express sin 285º as the function of an angle whose measure is less than 45º. What do we know? 285º in IV quadrant the sine of a IV quadrant angle is negative -sin 75º reference angle for 285º is (360 – 285) = 75º > 45º sine and cosine are co-functions complement of 75º is 15º < 45º sin 285º = = -cos 15º -sin 75º

  30. Trig Functions Using Radian Measures Algebraically: Find: sin (π/3) remember: π/3 radians π/3 60º sin 60º = ≈ .866… Using the calculator: Use the mode key: change setting from degrees to radians then hit: sin 2nd π ÷ ENTER 3 Display: .8660254083

  31. y 1 1 -1 unit circle 1 x -1 Un-unit circle  is any angle in standard position with (x, y) any point on the terminal side of  and r  1

  32. 4 r = 5 3 Model Problem (-3, 4) is a point on the terminal side of . Find the sine, cosine, and tangent of . Q II

  33. -1 r = 2 Model Problem is a point on the terminal side of . Find , the sine, cosine, and tangent of . Q III

  34. Model Problem Tan  = -5/4 and cos  > 0, find sin  and sec  When tangent is negative and cosine is positive angle is found in Q IV.

  35. Model Problem The terminal side of  is in quadrant I and lies on the line y = 6x. Find tan ; find . y = mx + b - slope intercept form of equation m = slope of line y = 6x m = 6 = tan  Q I

  36. Model Problem The terminal side of  is in quadrant IV and lies on the line 2x + 5y = 0. Find cos . y = mx + bslope intercept form of equation tan  = m = -2/5

  37. y 1 1 45º x -1 1 -1 Templates