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1. - 1. 1. - 1. Unit Circle. (A circle of radius 1.). (x, y). r =1. Every ray that comes from the origin and intersects the unit circle has a length of 1 (because it is a radius). y. x.

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  1. 1 - 1 1 - 1 Unit Circle (A circle of radius 1.) (x, y) r =1 Every ray that comes from the origin and intersects the unit circle has a length of 1 (because it is a radius) y x A right triangle can be created from every ray (excluding the axes) by adding a vertical line from the circle to the x-axis. (x, 0) The length of the side along the x-axis corresponds to the x-value of the ordered pair. The length of the side along the y-axis corresponds to the y-value of the ordered pair.

  2. Unit Circle

  3. a. Imagine a right triangle with a height of zero (Hyp) (Opp) (adj) Example 1 The x-value of the terminal side is -1, so The x-value of the terminal side is 0, so

  4. Example 2 V0 – initial velocity θ – measure of angle between ground and initial path of the ball g – acceleration due to gravity (9.8 m/s2) h = 40 h = 0 The possible maximum height of the ball is between 0 & 40 meters.

  5. Example 3 • Since the angle is > 90, find the reference angle. • Use Pythagorean Theorem to determine the values of the ordered pair of the intersection of the angle with the unit circle. 3. Since it’s a 45-45-90, it’s isosceles; both legs are the same; since the terminal side is in Quadrant 3, the x value is negative.

  6. Angles not on the Unit Circle (a good thing to memorize) Example 4 x = 5, y = -12, r = 13

  7. Example 5 θ is the angle formed at the origin. Since the terminal side is in Quadrant III, the x-value must be negative, so x = -3 Sin is opp over hyp so you know where to put each value. x = -3, y = -4, r = 5

  8. Signs of Trig functions: (another shortcut) All Students Take Calculus: STUDENTS only Sine/Cosecant are positive in Quadrant II ALL values are positive in Quadrant I CALCULUS only Cosine/Sec are positive in Quadrant IV TAKE only Tangent/Cotangent are positive in Quadrant III

  9. Hw: Page 298

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