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(0,1). ( –1,0). (1,0). (0,–1). Note 2: The Unit Circle. The unit circle has a radius of 1 with a centre of (0,0). P( x , y ). . (0,1). Let ’ s consider a blow-up of an angle in the first quadrant.

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  1. (0,1) ( –1,0) (1,0) (0,–1) Note 2: The Unit Circle The unit circle has a radius of 1 with a centre of (0,0) P(x, y) 

  2. (0,1) Let’s consider a blow-up of an angle in the first quadrant The radius from this angle  cuts the circle at point P whose coordinates are (x, y) 1 P(x,y) 1 y Label the points O and R, and drop perpendicular from P to the x-axis to make  OPR  R O x (1,0)

  3. (0,1) 1 P(x,y) 1 x = cos  y y = sin   R O x (1,0) Using the Triangle OPR, (cos, sin ) Therefore, the co-ordinates of P are..

  4. (0,1) 1 P(x,y) 1 y  R O x (1,0) By Pythagoras x2 + y2 = 12 So: cos2 + sin2 = 1

  5. We can use the unit circle to find values for cosΘ and sinΘ without the using the calculator Example: Use the Unit Circle to find the value of: sin 30° cos 45° Check on your calculator Sin 30° = 0.5 Cos 45° = 0.7071 ≈ 0.5 ≈ 0.7

  6. Exercise 11A Page 228

  7. STARTER: Use the Unit Circle to find the value of: sin 60° cos 72° sin 23° cos 36° ≈ 0.85 ≈ 0.3 ≈ 0.4 ≈ 0.8

  8. (0,1) ( –1,0) (1,0) (0,–1) This unit circle has Four QUADRANTS Q2 Q1  Q3 Q4 All angles are measured ANTICLOCKWISE from the positive x-axis

  9. x = cos y = sin  (0,1) ( –1,0) (1,0) (0,–1) Q1 angles.... x > 0 y > 0 As both x and y are positive cos > 0 sin  > 0

  10. x = cos y = sin  (0,1) x < 0 y > 0 ( –1,0) (1,0) (0,–1) Q2 angles.... As x is negative and y is positive cos < 0 sin  > 0 These are all the obtuse angles between 90 & 180. You may have noticed their cosines were negative

  11. 150° 30° Note 3: Obtuse Angles - Quadrant 2 Consider the points P and Q P = (cos 30, sin 30) Q = (cos 150, sin 150) Their y-coordinates are the same, so Q P sin 150 = sin 30 Their x-coordinates are opposite, so cos 150 = – cos 30 This applies to all Q1 & 2 angles related this way: cos (180 – ) = – cos  Formulae for angles in Quadrant 2 sin (180 – ) = sin 

  12. Exercise 11B Page 229

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