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Unformatted text preview: Math 1205 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x 2 + y 2 = 1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of a circle measured in the counterclockwise direction that subtends an arc length equal to 1 radius. Notice that the angle does not change with the radius. There are approximately 6 radius lengths around the circle. That is, one complete turn around the circle is 2 ! " 6.28 radians. Define the Sine and Cosine functions: Choose P ( x , y ) a point on the unit circle where the terminal side of ! intersects with the circle. Then cos ! = x and sin ! = y . We see that the Pythagorean Identity follows directly from these definitions: x 2 + y 2 = 1 (cos ! ) 2 + (sin ! ) 2 = 1 we know it as: sin 2 ! + cos 2 ! = 1 Example 1. Example 2. Determine: Determine: sin(90 ) and cos(90 ) sin(3 ! ) and cos(3 ! ) Recall that 90 corresponds to ! 2 radians . (How many degrees do 3 ! radians correspond to?) We can read the answers from the graphs: sin(90 ) = sin ! 2 " # $ % & = y coordinate of P = 1 sin(3 ! ) = sin(540 ) = y coordinate of P = cos(90 ) = cos ! 2 " # $ % & = x coordinate of P = cos(3 ! ) = cos(540 ) = x coordinate of P = " 1 Problems 1 and 2: 1. Locate the following angles on a unit circle and find their sine and cosine. a. ! 5 " 2 b. 5 ! 2 c. 360 d. ! " 2. Given: cos ! = 0 and sin ! = 1 . Find the following: a. the smallest positive ! that satisfies the given equalities. b. one other ! that satisfies the given equalities. Note: ! 1 and ! 2 should be in radians. There are six trigonometric functions. We have considered the sine and cosine functions. We can define the four remaining in terms of these functions. The tangent function: tan ! = sin ! cos ! The cotangent function: cot ! = cos ! sin ! The cosecant function: csc ! = 1 sin ! The secant function: sec ! = 1 cos ! We know all of the above functions will have points of discontinuity where the denominator is zero. The graphs of these functions all have vertical asymptotes at these points....
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 Spring '08
 FBHinkelmann
 Calculus, Unit Circle

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