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The Unit Circle

The Unit Circle. Part II (With Trig!!) MSpencer. 90°,. 270°,. Multiples of 90°,. 180°, . 360°, 2 . 0°, 0. 90°,. QII 90° <  < 180° <  < . Q I 0° <  < 90° 0 <  <. QIV 270° <  < 360° <  < 2. QIII 180° <  < 270°  <  <. 270°,.

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The Unit Circle

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  1. The Unit Circle Part II (With Trig!!) MSpencer

  2. 90°, 270°, Multiples of 90°, 180°,  360°, 2 0°, 0

  3. 90°, QII 90° <  < 180° <  <  Q I 0° <  < 90° 0 <  < QIV 270° <  < 360° <  < 2 QIII 180° <  < 270°  <  < 270°, The Quadrants (with Angles) 180°,  360°, 2 0°, 0

  4. r = 1 The Unit Circle Remember it is called a unit circle because the radius is one unit. So let’s add in ordered pairs to the unit circle.

  5. 90°, 270°, Multiples of 90°, (0, 1) r = 1 180°,  0°, 0 (1, 0) r = 1 r = 1 (1, 0) r = 1 (0, 1)

  6. 45°, 45° Notice that 45° or forms one of the two special right triangles from geometry. 45° 45°,

  7. 45° 45° 45°, Let’s review this triangle from geometry. Opposite the congruent, 45° angles are congruent sides. These sides are the legs of the right triangle. So the triangle is an isosceles right triangle.

  8. 45° s 45° The hypotenuse is the length of either leg, s, times ; thus, s . s 45°, Let’s call the two congruent legs s.

  9. 45° s 45° s 45°, Lastly, now remember that the hypotenuse is the radius of the unit circle, which means it must equal one. Solve for s.

  10. 45°, 45° 45° 45°, 1 The distance across the bottom side of the triangle represents the x-coordinate while the right, vertical side represent y.

  11. 90°, 270°, Signs and Quadrants The signs of each ordered pair follow the signs of x and y for each quadrant. Q II (, +) Q I (+, +) 180°,  0°, 0 Q III (, ) Q IV (+, )

  12. 135°, 45°, 45° 45° 45° 45° 45° 45° 225°, 315°, Multiples of 45°,

  13. 60°, Notice that 60° or forms the other special right triangle from geometry. 30° 60° 60°,

  14. 30° 60° The medium side opposite 60° is times the smallest side, or . 60°, Let’s review this triangle from geometry. Call the the smallest side opposite 30° s. 2s The hypotenuse is twice the smallest side, or 2s. s

  15. 30° 60° The medium side opposite 60° is 60°, The hypotenuse is the radius of the unit circle, which means it must equal one. Solve for s. 2s = 1 s

  16. 60°, y x 60°, Notice that since the triangle is taller than it is wide, that the y-coordinate is larger than the x-coordinate.

  17. 120°, 60°, 240°, 300°, Multiples of 60°,

  18. 30°, 60° y 30° x 30°, Notice this is the same special right triangle as for 60° except the x and y coordinates are switched.

  19. 150°, 30°, 60° 30° 210°, 330°, Multiples of 30°,

  20. Ordered Pairs and Trig From geometry, recall SOHCAHTOA, which defines sine, cosine, and tangent. sine (Sin) = cosine(Cos) = tangent (Tan) =

  21. 30°, 60° 30° Ordered Pairs and Trig Cos 30° = cos 30° = Notice that the cosine of the angle is simply the x-coordinate!

  22. 30°, 60° 30° Ordered Pairs and Trig Sin 30° = sin 30° = Notice that the sine of the angle is simply the y-coordinate!

  23. Ordered Pairs: Cosine & Sine (cos , sin ) (x, y)  And this is true for ANY angle, often called . cos = x sin  = y

  24. 90°, 270°, Signs for Cosine and Sine The “signs” of cosine and “sine” follow the signs of x and y in each quadrant. Q II (, +) Q I (+, +) 180°,  0°, 0 Q III (, ) Q IV (+, ) So in QII, for instance, cosine is negative while sine is positive.

  25. (0, 1) 90°, 150°, 30°, 120°, 60°, 135°, 45°, (1, 0) 180°,  (1, 0) 0°, 0 225°, 315°, 240°, 300°, 210°, 330°, 270°, (0, 1) The Whole Unit Circle Together (Grouped)

  26. (0, 1) 90°, 150°, 30°, 120°, 60°, 135°, 45°, (1, 0) 180°,  (1, 0) 0°, 0 225°, 315°, 240°, 300°, 210°, 330°, 270°, (0, 1) The Whole Unit Circle Together (In Ascending Order)

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