v. Trigonom.etry, part 1

Transcription

1 v. Trigonom.etry, part A. Angle measurement ya\ x The standard position for angles in the xy-plane is with the initialside on the positive x-axis and the counterclockwise direction taken to be positive. Two common units for measuring angles are dee;reesand radians. 90. Degrees 360. = revolution, so. = -.L th of a revolution. 360 Radians. One radian is defined to be the angle subtended at the center by an arc of length r on a circleof radius r. The circumference of a circleof radius r has length '2Tr, so r units can be marked off.. 2T times". n other words, revolution = 2TT radians. r 9' = radian 9'= ~ radians Relation between degrees and radians: Since revolution = 3600 = 2TTradians, This gives the conversion relations: 800 = TTradians. To convertdegrees to radians, multiply by TTradians o 80 e.g. 450 = 45.. Tradians = Tradians V.l

2 . 0 To convert radians to degrees, multiply by TTra.8i lans ' o. e.g. - J!. radians = - TTradians TTtadians = n calculus, radian measure is used exclusively (because it simplifies the differentiation formulas for the trig. functions). Examples: some angles in standard position br - tt rad 2 hyp~tenuse pposi te B. Jrig. functions for anges in right triangles e adjacent -. f e is an acute angle (more than 0 but less than 90 ), the trigonometric functions of e can be defined as ratios of sides in a right triangle having e as one angle. The six possible ratios give the six trig. functions. opposite. sine of e = sinee) = hypotenuse' cosecan t of e = csc (e) =.hyr. opp cosine of e = cos(e) = ad.iacent. hypotenuse J secant of ~ = sec(e) = ~ adj tangent of e = tan (e) = op?osite ; cotangent of e = cot (a) = ad.i adjacent opp Notes: We usually write sin a instead of since) J etc. Cotangent of a is sometimes abbreviated ctn(e). Remember to think" sine of a", not" sine times a"!! V.2

3 2. Reciprocal relations: csc & = sin e and""" sin & "= - CSC& sec e = cose and COS&= sece cot & = tane and tan e = --.. cote Also note that tan & = sin & COS& and cot & =. cos &,sin &. 3. The most common tr~: " " 2A5.. sin 45 = cos 45. tan 45 = " sin 60. = cos 30. = 3 2. sin30- = cos60. = For other angles in right triangles, consult trig. tables or use a calculator. (Take care, however -- be sure you and your calculator agree on whether you're punching in degree measure or radian measure!) c. Dg. functi.qns.jor general ang~ x. Place the angle e in standard position. Mark a point P: (x,y) on its terminal side. Let r be the distance from the origin to P. Then the six trig. functions of e are defined by: V.3

4 sin & = y. r cos & = X. r. tan & = y. x csc & = :.; sec & = L; cot & = K. Y x Y provided the ratio in question is defined (does not have zero in the denominator). Notes: When & is an acute angle, this definition gives the same results as the definition in Section B. f P is chosen so that r =, the formulas simplify; cos & and sin & are the x and y coordinates of the appropriate point on the unit circle x2 + y2 =. 2. Examp~ (-,) cos 3T = - '2 4-2 or ~ 2 sin 3 TT= }2 r;: or -2 tan 3T = - (-,0) T costt= -- = - - SnTT=. 0- = 0 tan TT= JL = 0 - but csc TT, which would be.., o. is undefined. 3. Trig. functions at the popular ang~ 0 TT T!. TT 3TT 6 TT Sn 0.. i cos J3 i tan 0 J3 3 undef'd 0 undef'd 3 V.4

5 l 4. Rgference ang~. sin + cos, tan - tan + sin, cos - all + f & is not a quadrantal angle (t~e terminal side is not an axis) " then Trig. fn. of & = :t.same trig. fn. of.reference cos +. angle for &, sin, tan - and the choice of.:t is determined by the chart at left. The. reference angle for &.. is the smallest unsigned angle between the terminal side of & and the x-axis. Examp~: & = 3u 4 t. reference angle = : sin 3TT= + sin TT = cos 3TT= -cos. = - L. 4 4 r:' 4 4 r;:' ~2. ~2 3TT TT tan - = - tan - = reiference angle =.... TT =J3. \, 6 Sn - ~ = -sn '6 = - 2". cos 6 - +cos 6 2' J TT tan - ~ = - tan '6 = - J3' The trig functions of TT are the same 6 as those of - TT, computed above. 6. This method is based on the following identities, which come from the definitions of the trig. functions: sin (-&) = - sin e ; cos (-&) = cos a ; sin(&+tt)= -sin e ; COS(&+TT)= - cos e. Also, note that the trig. functions are periodic: they repeat every 2TT radians: sin(e+2tt) = sin e; cos(e+2tt) = cos a, etc. See the next handout, "Trigonometry, part 2, plus conic sections", for other trig. identities. V.S

6 Exercises V ABQ. Convert to radians.. (a) 50 (b) Convert to degrees. (a) 5rr/6 radians. (b) - 5rr/2 radians 3. For each of the following angles, find sin e, cos e, tan e. Use reference angles where appropriate... (a) 5rr/4 (b) -rr/6 (c) 3rr (d) Find csc(4rr/3), see (4rr/3), cot (4TT/3). 5. Given: :::J3 4 Find (a) sin e (b) cos & (c) tan & 6. Given:~ 5 Find (a) (b) (c) (d) the length sin e of the third side COS& tan e 7. n the diagram at right, find the radian measure of &. 8..~..'.'... };: '\.~~: ~ mt =5280 ft. 3 An airplane is flying over a house mile from the spot where you're standing. At that instant, the plane's angle of elevation from your viewpoint is TT/6 radians. Find the plane's altitude (in feet). V.6

7 Answers to Exercises V.(a) tt/2 (b) 5rr/2 2.(a) 500 (b) (a) -/ ff, -/ ff, (c) 0, -, /ff, - 2, /ff 5.(a) 3/5 (b) 4/5 (c) 3/4 6.(a) (side) = 82 so (side) =./64-25 = /39 (b) 5/8 (c) /39/8 (d) 5/ /39 or 5/39 /39 7. By ratios: & is to angle of whole circle as 3 is to circumference, & 3 3 or - =, so & = -. 2rr 2rr.2 2 (b) /2,../372, /?) (d) -/372, /2, -ff 8. tan T= alt.,so alt. = ft. = 760J3 ft. 6 mi. 3.. V.7