Chapter 5.3: Circular Trigonometric Functions

Transcription

1 Chapte 5.3: Cicula Tigonometic Functions A efeence tiangle is fomed b dopping a pependicula (altitude) fom the teminal a of a standad position angle to the -ais, that is, again, the -ais. The efeence angle will be the positive, acute angle of the efeence tiangle between the teminal a and the -ais. Refeence tiangles ae used to find tigonometic values fo thei standad position angles. The ae of paticula impotance fo standad position angles whose teminal sides eside in Quadants II, III, o IV. Eample 1: Daw a efeence tiangle fo an angle that teminates in the following quadants. Label the efeence angle and the efeence tiangle. Descibe mathematicall how to find the efeence angle in each case in tems of both degees and adians. (a) Quadant I (b) Quadant II (c) Quadant III (d) Quadant IV A tigonometic function is a atio of 2 of 3 sides of a ight tiangle fomed b dawing a efeence tiangle with efeence angle ef fom an independent angle in standad position. Eample 2: Daw a efeence tiangle in Quadant I, dopping ou pependicula fom the point, a. Label the hpotenuse, then list all the possible atios of,, and. on the teminal Page 1 of 6

2 Let be an eal angle, and let, be the teminal point fom which the pependicula is dopped ceating a efeence tiangle with hpotenuse. Then we define the si atios of the side lengths of the efeence tiangle to be the following sin cos tan (sine function) (cosine function) (tangent function) csc sec cot (cosecant function) (secant function) (cotangent function) Because these functions can be defined b otating an adius though an angle in standad position, the ae efeed to as cicula tigonometic functions. Eample 3: 5 If sin and , find the simplified, eact value of the othe five tig functions of. Find 6 the value of and ef using the calculato. Depending on which quadant an angle teminates, the sign of each of the si tig functions can be eithe positive o negative. Because will end up being the adius of otation, it is alwas positive. Theefoe the signs of the tig functions ae detemined eclusivel b the signs of and. The chat at ight show these signs. Page 2 of 6

3 Eample 4: 5 If tan and csc 0, detemine the simplified, eact value of the othe five tig functions of. 12 Find the value of 0,360 and ef. Eample 5: If cot 2 and cos 0, detemine the simplified, eact value of the othe five tig functions of. Find the value of 0,360 and ef. Eample 6: If the teminal side of passes though the point 4,3, find the simplified, eact values of all si tig functions of. Page 3 of 6

4 A quadantal angle is an angle that teminates on eithe the - o - ais. Quadantal angles have no elevant efeence angles since in each case the efeence tiangle eithe collapses veticall o hoizontall. Eample 7: Fo a cicle of adius 1 unit centeed at the oigin, find the value of the si tig functions fo each of the following quadantal angles: (a) 0 (b) 90 (c) 180 (d) 270 (e) 360 (f) 720 (e) 1080 The cicle mentioned in the pevious eample is called a Unit Cicle. Refeence angles of 30, 45, and 60 show up quite often in calculations. Consequentl, it is woth developing the cosine and sine values fo all the angles within one positive otation aound the Unit Cicle. This will be LOTS of FUN! Befoe we can do that, though, we must eview two special tiangles fom geomet. Eample 8: Daw a and a tiangle. Fo each, scale the hpotenuse to be one unit long. Then find the following: (a) cos30 (b) sin 30 (c) cos45 (d) sin 45 (e) cos60 (f) sin 60 Page 4 of 6

5 We will now develop the Unit Cicle Each coodinate, on the unit cicle not onl epesents a point on the cicumfeence of the cicle, but, moe impotantl, epesents the cosine and sine values, epectivel of the angle in standad position. That is, cos,sin Fom onl these two tig functions, we can obtain the othe fou b using the following tigonometic identities (An identit is an equation that is tue fo all values of the vaiable in the domain of each epession.) sin tan cos cos cot sin 1 sec cos 1 csc sin Page 5 of 6

6 *Note: These Unit Cicle atios wok egadless of the size of the cicle o tiangle. Since the scale facto affects all thee sides, it will alwas divide out in the atios. Eample 9: Using the Unit Cicle and the identities, find the si tig functions fo the following angles (a) (b) (c) (d) Theoem Coteminal angles have the same tig atios. Eample 10: Find the simplified, eact value of the following using the Unit Cicle. 16 (a) sin 210 (b) tan 3 (c) cos205,155 (d) csc3655 Not all angles ae on the Unit Cicle. Fo these angles, if we want to appoimate thei tig atios, we must use the calculato eclusivel. Eample 11: Evaluate the following on the calculato. Be sue ou ae in the coect mode on ou calculato. Repot thee decimals. (a) sin '12" (b) sec 45 (c) cot (c) 19 csc 5 Note: Although we have been measuing angles in both degees and adians, fom this point fowad, we will be using adians, since these ae eal numbes and unitless. Page 6 of 6