Section 5.2: Trigonometric Functions of Angles

Transcription

1 Section 5.: Tigonometic Functions of Angles Objectives Upon completion of this lesson, ou will be able to: Find the values of the si tigonometic functions fo an angle θ with given conditions. Given an angle and the length of one side of a ight tiangle, use the tigonometic functions to find the length of an missing side. Solve application poblems given an angle and one side of a ight tiangle using the tigonometic functions. Know the eact values of the tigonometic functions fo 30, 45, and 60 and the quadantal angles. [Also the adian fom.] Using a calculato, find the tigonometic function of an given angle. Find the quadant containing an angle θ with given conditions. Use the ecipocal, tangent and cotangent, and the Pthagoean identities to simplif epessions, veif identities, o wite tigonometic epessions in tems of a diffeent tigonometic function. secθ = 1 cscθ = 1 cotθ = 1 tanθ tanθ = cotθ = cos θ + sin θ = 1 1+ tan θ = sec θ cot θ + 1 = csc θ Requied Reading Swokowski/Cole: Section 5., pages Discussion Below is a ight tiangle on a coodinate gid in a cicle of adius. The cicle and the tiangle have the point P(, ) in common. Also pictued is the same ight tiangle without the gid o cicle.

2 P(, ) otenuse osite side θ θ acent side The si tigonometic functions; sine, cosine, tangent, cotangent, secant, and cosecant; ae defined as atios of the lengths of the sides of a ight tiangle. Tigonometic Ratios/Functions Using -, -notation fom the figue above, the si tigonometic atios o functions ae defined as: osite 1 sin θ = = csc θ = = otenuse = = acent otenuse 1 secθ = = osite 1 tan θ = = = cotθ = = = acent tanθ Note the elationships between the tigonometic atios: Cosecant is the ecipocal of sine. Secant is the ecipocal of cosine. Tangent is the atio of sine divided b cosine. Cotangent is the ecipocal of tangent, which is also cosine divided b sine.

3 Notation: The common labeling of ight tiangles is to identif the ight angle as angle γ with the two acute angles labeled α and β. The otenuse, the side osite the ight angle γ, shall be labeled c. The side osite angle α β shall be labeled a, and the side osite angle β shall be labeled b. a c γ b α Eample 1: Fo the given ight tiangle, find the 6 tig functions fo angle α. Solution: β a = m c = m Befoe setting up the atios, it is necessa to find the length of side b. This is accomplished b using the Pthagoean Theoem. γ b α a + b = c + b = b = 5 sinα = = cosα = = tanα = = b = ± Choose +m since b epesents a length. csc α = sec α = cot α = Now find the same tigonometic functions fo angle β. sin β = = cosβ = = tan β = = csc β = sec β = cot β = 3

4 Notice that the definition of the tigonometic function emains constant egadless of the angle used; howeve, the osite and acent sides ae edefined elative to the chosen angle. Also notice that sin α = cos β and that sin β = cos α. This is due to the fact that α and β ae complementa angles and we ae dealing with cofunctions. The topic of cofunction will be coveed in section 6.3. It is helpful to set up the si tigonometic functions in table fom as in the above eample fo eas computation. An epedient method to use when woking with man poblems in tigonomet is to daw a pictue of the situation. It is citical to daw the pictue coectl, especiall when locating angles in a paticula quadant to constuct a tiangle. The tet summaizes the signs of the tigonometic functions on page 34. It helps to emembe that the cosine function is diectl elated to the -coodinate and will theefoe have the same sign as the -coodinate will have when plotting points in the fou quadants. The same holds tue fo the sine function, it is diectl elated to the -coodinate. Since each function shaes the same sign as its ecipocal function, i.e. sine and cosecant, cosine and secant, and tangent and cotangent, b emembeing the signs fo sine and cosine in the fou quadants and that tangent is the atio of sine divided b cosine, ou will automaticall know the coesponding sign fo secant, cosecant, and cotangent. Instead of using the Pthagoean identit sin θ + cos θ = 1, the poblems can also be done b dawing a pictue of the situation. Then fom the pictue, it is a matte of using the tigonometic definitions and eading the values fom the pictue. The Pthagoean Identities The equation of the cicle pictued at the top of page is + =. We know b the definition fo sine and cosine that cos θ = o = and sin θ = o =. Substituting into the cicle equation, we have cos θ + sin θ =. Since 0, we can divide b, and we have the Pthagoean identit sin B dividing sin B dividing sin θ + cos θ = 1 b sin θ, we get 1 + cot θ = csc θ. θ + cos θ = 1 b cos θ, we get tan θ + 1 = sec θ. Tigonometic Identities θ + cos θ = 1. Establishing an identit is basicall poving that a tigonometic equation is tue fo all values of the agument. You ma wok on onl one side of the identit to achieve the othe given side. Even though ou ma not wok on both sides, ou ma cetainl look to see what ou ae ting to pove which ma povide ou with clues as to the net step in eaching ou goal. 4

5 Begin b woking on the moe difficult side. Convet to the sine and cosine functions if ou do not see anothe technique to use. Be familia with the basic identities listed on page 377; these ae the main tools fo veifing othe, moe comple identities. You will want to memoize these since the will NOT be povided on the fomula sheet fo the test o the final. Be Patient and Pesistent!!! If things ae not woking out, sometimes it is necessa to stat ove with a diffeent plan, o begin again b woking on the othe side. Eample on page 374 shows the deivation fo the 30, 45, and 60 angles. A chat summaizing the findings is at the top of page 375. The soone ou familiaize ouself with these values, the easie ou will find this couse. Eample : Evaluate cos 45 ( sin 30 tan 60 ) Solution: cos 45 ( sin 30 tan 60 ) 1 = ( ) 1 = 3 5 = 3. = 5 4 Pa close attention to ou use of notation. ( tanθ ) = tan θ tanθ Please pint the file of the Unit Cicle which povides a visual of the mateial intoduced in section 5.. 5

6 Man applications of ight tiangles involve the angle of elevation and the angle of depession. The angle of elevation (see Figue 1) efes to the angle measued fom the hoizontal upwad to the object in question. The angle of depession (see Figue ) efes to the angle measued fom the hoizontal downwad to the object. It is tpical that the angle of depession is eteio to the ight tiangle depicting the application. object obseve hoizontal angle of depesession angle of elevation obseve hoizontal Figue 1 Figue object Pactice Poblems Wok these poblems. Answes to the odd numbeed poblems can be found at the end of ou tet. Section Pages Eecises , 5, 7, 9, 11, 13,, 19, 1, 3, 5, 9, 31, 33, 39, 41, 43, 45, 53, 55, 57, 59, 61, 63, 65, 71, 73, 1, 3, 5, 9, 91 6