PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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This 15 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 10 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.
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Date Created: 09/19/15
L23 Complementary Angles Cofunctions Trigonometric Functions of Angles g Trigonometric Functions of General Angles Two acute angles are called complementary if their sum is the right angle Example Find the complementary angles for 27 it a 50 b c 7 6 Note In a right triangle a 90 therefore a and are complementary Also b a b sm cosa cos sma tan cota c c a c c a csc gseca sec csca cot gtana a The functions sine amp cosine tangent amp cotangent and secant amp cosecant are called cofunctions of each other 268 Complementary Angles Theorem Cofunctions of complementary angles are equal I Note The angles 6 and 90 6 are complementary therefore the theorem s statement can be written as Function 6 C0functl390n 90 6 Or Function 6 C0functl390n 6 Example Use the Complementary Angle Theorem to fill in the blank spaces sin10 cos c0s30 sin 7 7 tan 2 cot csc 2 sec 4 6 Example Find the exact values of the expressions tan 56 cot34 57 7 sm cos 2 12 12 269 The Trigonometric Functions of Angles g g g 61450 4 7r 5111 4 7r 0057 4 7r CSC 4 7r seci 4 cotz 4 Example Find the exact value of the expression escaj cot 91 3 7r s1n 3 60 7r 7 7 cos tan cot 3 3 3 Note By using the Complementary Angles Theorem 7f 7 8111 COS 7z 7 7r 7 cos s1n tan cot 6 3 6 3 6 3 9130o 9145o 6160quot 6 4 3 sin 6 cos 6 tan 6 cot 6 Note The entries in the table above must be memorized Example Find the exact values of the expressions tang emf 3 3 7239 7239 sec csc 4 esc1 00tl 272 Trigonometric Functions of General Angles Let P xy be apoint on the terminal side of 6 and 0 dPOr rgt0 0 Thus r I le yZ Wede ne cos6 sec6 x 0 r x s1n6x csc6 y 0 V y tan6x x 0 cot6 y 0 5 y The ratios are independent on the choice of a point P xy for instance cos6 f i where R xPyl r n is any other point on the tenninal side rl J06Z y1Z they are functions only of angle 6 Thus The six ratios above are called the trigonometric functions of angle For an acute ang e 19 GOSH f adjacent r hypotenuse 5mg X r tang X x Example Find the values of the remaining trigonometric functions if c0519 7 and I9 is in Quadrant III The Values of the Trigonometric Functions of the Quadrantal Angles For each of the quadrantal angles 6 we choose the point P x y on the terminal side of 6 6000 lt gt P10 61900 lt gt P01 a 7r180 lt gt P 710 6 37 270 gt P 071 Since r l and the point on the terminal side of 6 is P xy then c056x sin6y tan6l cot6 x y x150 yiO 7r 37r 6000 6 90 6 1800 6 270 2 2 c056 sin 9 tan 9 cot 9 Coterminal Angles Two angles in standard position are called coterminal if they have the same terminal side by a point on the terminal side thus they are equal for coterminal angles The values of a trigonometric function are defined cos 9 2k coslt9 sec 9 2k sec 9 sin6 2k sin 9 csc 9 2k csc 9 tan 9 2k tan 9 cot 9 2k cot 9 Note An angle coterminal with 6 is 6 27rk or 6 360 k Where k is an integer number Example Find the angle in the interval 0 360 that is coterm inal with Example Find the angle coterm inal with 6 157 which lies a in interval 0 27 b in interval 77 7 Example Find the exact value of each of the following sin 720 23 sec 7 6 llIr s1n 2 cotll37r The Sigis of the Trigonometric Functions Let P xy be a point on the terminal side of 6 and dP0 r 1 The signs of the trigonometric functions of an angle 6 determined by the quadrant where 6 lies and can be easily obtained from the siX ratios for the trigonometric functions assuming that r l sin6y cos6x tan6Z x 05061 sec6l cot6 y x y Example Name the quadrant where the angle 6 lies if sin6lt 0 and tan6 gt 0 Reference Angle Let 8 be an angle that lies in a quadrant The reference angle for 8 is the acute angle formed by the terminal side f 8 and the xaxis Li 7i Example Find the reference angle a for 8 if 9092 b875 083300 nding the Exact Values of a Trigonometric Function by using the Reference Angle Let a be the reference angle for a nonquadrantal angle 6 and Pi iapoint on the terminal side of 6 r 2 Note a is the acute angle in the right triangle with the opposite lyl adjacent M and hypotenuse r sin6X gt Isin msina gt sin6sina r r cos6i gt cos67icosa gt cos6cosa r r tan X gt tan6tana gt tan6tana x x In order to find the Value of a trigonometric function of a nonquadrantal angle 6 we do the following mine the quadrant where 6 lies 2 Find the reference an le 1 Value of the function at 6 is the positiv negative depending on the quadrant Value ofthe function at I Example Find the exact values of 7239 S111 4 297 tan 6 csc 225 281 Finding Values of the Remaining Functions by Using the Reference Angle Example Use the reference angle a to find the values of the rema1n1ng tr1gonometrlc functlons at 6 1f s1n 6 2 and tan6lt0 1 Graph the angle 6 2 Find the reference angle a 3 Draw a right triangle with the acute angle a 4 Find the values of remaining functions of the angle 6 evaluating the corresponding ratios in the right triangle and adjusting the Sign according to the quadrant where 6 lies 282
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