ࡱ> BDA}'` YbjbjLULU 3p.?.?!UdNdNdN8N O1DOHQ(pQpQpQTTTdddde?j?R$uhvu@V'TT@V@VvpQpQ0^0^0^@VpQpQd0^@Vd0^0^0^pQO dNXh0^,_010^s([s0^s0^TZU@0^[U4UTTTvv]XTTT1@V@V@V@V2 :D : HOMEWORK: Practice completing the entire unit circle with angles in radian measure with their corresponding (x, y) coordinates. Sec 2.5: 1, 9-14 Complete the Unit Circle Worksheet Complete remaining examples from the lecture notes Read Sec 2.4. UNIT CIRCLE: A point (x, y) is on the unit circle (circle with radius 1) if  EMBED Equation.3  If we draw our common angles on the unit circle we get If we put these all together and use symmetry, we get the UNIT CIRCLE! It is EXTREMELY IMPORTANT that you memorize and know the exact values of all details on the unit circle.  HYPERLINK "http://upload.wikimedia.org/wikipedia/commons/c/c9/Unit_circle_angles.svg"  INCLUDEPICTURE "http://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Unit_circle_angles.svg/600px-Unit_circle_angles.svg.png" \* MERGEFORMATINET  THEOREM: If  is an angle in standard position and P=(a, b) is a point on the terminal side of  that is on the circle x2+y2=r2 sin( = b / r cos ( = a / r csc ( = r / b (b ( 0) sec ( = r / a (a ( 0) tan ( = b / a (a ( 0) cot ( = a / b (b ( 0)   *** Unit Circle Applications: To be on the unit circle, the radius MUST be equal to 1! Let t be a real number and (x, y) be a point on the unit circle. (NOTE: this means  EMBED Equation.3  and r = 1) sin t = y/1 = y csc t = 1 / y (y ( 0) cos t = x/1 = x sec t = 1 / x ( x ( 0) tan t = y / x (x ( 0) cot t = x / y (y ( 0) Note: The unit circle is just a special case of this general theorem! Example: A) Find the values of the trig functions corresponding to (-4/5, 3/5) sin  = csc  = cos  = sec  = tan  = cot  = NOTE: If  is on the unit circle, then r = 1 so sin=b, cos = a, and tan=b/a! THIS IS ONLY TRUE on the unit circle! If the point is not on the unit circle, then r `" 1 so we must use the general Theorem above. We will see examples of this in section 2.4. In this case we must find r using the Pythagorean theorem. If you are unsure if a point is on the unit circle, you can use the pythag. Thm. to check and see if it satisfies  EMBED Equation.3  B) Assume that (a, b) are the coordinates of a point on the unit circle at the angle . Find the coordinates of P( +( ) = P(( - ) = P(-) = C) Find the values of the following trig functions: sin((/4) cos(2(/3) tan((/6) csc(4(/3) sec(11(/6) D) Give the coordinates of 5(/6 5(/4 3(/2 SPECIAL VALUES of Trig functions: You MUST have the values of all of the trig functions at the common angles MEMORIZED! anglesin tcos ttan t sec tcsc t cot t0(/6(/4(/3  (/22(/33( /45(/6( 7(/65( /44(/33( /25(/37( /411(/62 ( Signs of the trig functions in various quadrants: In general,  (-, +) (+, +) sin( and csc ( >0 sin( and csc ( >0 cos(, sec( <0 cos(, sec( > tan(, cot( <0 tan(, cot( >0  (-, -) (+, -) sin( and csc ( < 0 sin( and csc ( < 0 cos(, sec( < 0 cos(, sec( > 0 tan(, cot( > 0 tan(, cot( <0 PROPERTIES: FUNCTION DOMAIN RANGE sin, cos all reals [-1, 1] tan, sec all reals except (/2 + k (k integer) cot, csc all reals except (k (k an integer) | sin t | < 1; | cos t | < 1; | sec t | > 1; | csc t | > 1 RECIPROCAL identities: sec t = 1 / cos t csc t = 1 / sin t cot t = 1 / tan t Periodic Functions: A function if said to be periodic if  EMBED Equation  for all  in the domain of f. 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