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

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Unit Circle

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

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

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Example 3 1.Since the angle is > 90, find the reference angle. 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. 2.Use Pythagorean Theorem to determine the values of the ordered pair of the intersection of the angle with the unit circle.

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

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

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

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Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle

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