[PDF] 146 Chapter 13. The Trigonometric Functions (LECTURE NOTES 9)

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146Chapter 13. The Trigonometric Functions (LECTURE NOTES 9)

Chapter 13The Trigonometric FunctionsWe look at trigonometric functions and their derivatives.13.1 Definitions of the Trigonometric FunctionsConsider an angle with origin (vertex) at the origin of the coordinatesystem and

tworayswhere theinitial sideray is along thex-axis andterminal sideray is at the end of a rotation of angleθ.Acute,right,obtuseandstraightangles occur when 0 o< θ <90o,θ= 90o, 90o< θ <180oandθ= 180orespectively. Forradius,rand arc (length),s, of a circle,radian measureofθis defined ass r; where, notice, if radius of circle is one (1), aunitcircle, radian measure equals arc lengths. An angle can be measured in eitherdegreesorradians, where

1 radian =

180o

π,1o=π180radians.

Letrbe distance from origin to point (x,y) on terminal side ray. Then sinθ=y rcscθ=ry,(y?= 0) cosθ=x rsecθ=rx,(x?= 0) tanθ=y x,(x?= 0) cotθ=xy,(y?= 0) where notationtorxcan be used instead ofθ; for example, sintor sinxcould be used instead of sinθ. Also, some trigonometric identities are: cscθ=1 sinθsecθ=1cosθcotθ=1tanθ tanθ=sinθ cosθcotθ=cosθsinθsin2θ+ cos2θ= 1 147

148Chapter 13. The Trigonometric Functions (LECTURE NOTES 9)

Values for trigonometric functions are typically found using a calculator but some values can be found for triangles with special angles given in the Figure. For example, for 30 o-60o-90otriangle, where 60o= 60·?π

180?=π3, so sin60o= sinπ3=yr=⎷

3 2. xy60 o (a) 30 - 60 - 90 triangle(b) 45 - 45 - 90 triangle ooo oo 30
90
0 xy45 45
90
oooo o oy = 3 r = 2 x =1 1 _ r = 2 1_ x = 1 y = 1 0 Figure 13.1 (Trigonometric functions and special angles) A trigonometric function isperiodicbecause it is a functiony=f(x) with real number asuch thatf(x) =f(x+a) for allx;smallesta, when the function repeats itself, is theperiodof the function. Periods of both sinxand cosxare 2π; theiramplitudes (half their range ("height") from -1 to 1) are both 1. Furthermore, constantsa,b,c,d transform graphs of bothasin(bx+c)+dandacos(bx+c)+din the following ways: •amplitudeaincreases (decreases) "height" of graph for|a|large (small) whena <0, graphs reflected inx-axis ("flipped"),

•constantb(assumeb >0) affects period;

for example, graph ofy= sin(bx) looks likey= sinxbut with periodT=2π bif 0< b <1, period completed more slowly (longer period) than whenb= 1 ifb >1, period completed more rapidly (shorter period) than whenb= 1 •horizontal shiftcmoves graphsleft(c >0) orright(c <0) •vertical shiftdmoves graphs up (d >0) or down (d <0)

Phase shift,c

b, gives number of units sinbxor cosbxare shifted horizontally; for example, ifc= 2π,b= 1, then sin(x+ 2π) isc b=2π1= 2πunitsleftof sinx, whereas ifc= 2π,b= 2, then sin(2x+ 2π) is2π

2=πunitsleftof sin2x.

Exercise 13.1 (Definitions of the Trigonometric Functions)

1.Introduction to angles, radians and trigonometric functions.

Section 1. Definitions of the Trigonometric Functions (LECTURE NOTES 9)149 (1,0)xy t = /3(x,y) = (cos , sin ) = (cos 60, sin 60 ) x + y = 122unit circle xy positive (counterclockwise) angle (or rotation) negative (clockwise) angle (or rotation)60 o -130 oterminal side initial side terminal side (a) angles in degrees(b) angles in radians60 o t = -13 /18 -130o πo o/3

π/3π

(x,y) = (cos , sin ) = (cos -130, sin -130 )o o -13 /18π-13 /18π yx xy r r

30 - 60 - 90 triangleo oo

Figure 13.2 (Angles, radians and trigonometric functions) (a) (i)True(ii)False. Origin vertex, and two rays, initial side and terminal side, form anangle. (b) Consider Figure (a). Rotating ray is called (i)terminal side(ii)initial side, whereas positive half of thex-axis is calledinitial side. (c) Counterclockwise rotation of 60 ois (i)positive(ii)negativerotation, whereas clockwise rotation of-130ois a negative rotation. (d) Angle of 90 oequals (i)-270o(ii)270o(iii)-180o.

Hint: 270-180 = 90.

(e) 60 oangle in figure (b). i. Angle 60 otracesarc length(i)-13π

18(ii)π3radians alongunitcircle.

ii. Horizontal distancex= cos60o= (i)0.5(ii)0.87

Calculator: MODE DEGREE ENTER COS 60 ENTER

iii. Vertical distancey= sin60o≈(i)0.5(ii)0.87

Calculator: SIN 60 ENTER

iv. Horizontal distancex= cosπ

3= (i)0.5(ii)0.87

Calculator: MODE RADIAN ENTER COS 2ndπ/3 ENTER

v. Vertical distancex= sinπ

3= (i)0.5(ii)0.87

Calculator: SIN 2ndπ/3 ENTER

vi. (i)True(ii)FalseSince cos60o= cosπ

3and sin60o= sinπ3point

(x,y) can be determined by evaluating trigonometric functions using eitherdegrees or radians. vii. Since this is 30 o-60o-90otriangle wherex= 1,y=⎷

3 andr= 2,

cos60 o= cosπ

3=xr=12= (i)0.87(ii)0.5see special triangle figure

sin60 o= sinπ

3=yr=⎷

3

2≈(i)0.87(ii)0.5see special triangle figure

Notice 30

o-60o-90otriangle assumes radiusr= 2, not 1, as given in figure (b).

Ifradius was specified as 1,r=2

2= 1, thenx=12andy=⎷

3 2.

150Chapter 13. The Trigonometric Functions (LECTURE NOTES 9)

(f)-130oangle in figure (b). i. Angle-130otraces arc length (i)π

3(ii)-13π18radians.

ii. Horizontal distancex= cos(-130o) = (i)-0.64(ii)-0.77

Calculator: MODE DEGREE ENTER COS -130 ENTER

iii. Vertical distancey= sin(-130o)≈(i)-0.64(ii)-0.77

Calculator: SIN -130 ENTER

iv. Horizontal distancex= cos?-13π

18?= (i)-0.64(ii)-0.77

Calculator: MODE RADIAN ENTER COS 2nd-13π

18ENTER

v. Vertical distancex= sin?-13π

18?= (i)-0.64(ii)-0.77

Calculator: SIN 2nd-13π

18ENTER

vi. (i)True(ii)False

Since cos(-130o) = cos?-13π

18?and sin(-130o) = sin?-13π18?point

(x,y) can be determined by evaluating trigonometric functions using eitherdegrees or radians. vii. (i)True(ii)False This case doesnotinvolve either the 30o-60o-90otriangle or the 45
o-45o-90otriangle and so neither can be used to determine the cosθor sinθfunctions unlike for the 60ocase.

2.Converting Degrees To Radians.

1 o=π

180radians

(a) Angle of 180 oequivalent to 180×π

180= (i)π2(ii)π(iii)3π2radians.

(b) Angle of 90 oequivalent to 90×π

180= (i)π2(ii)π(ii)3π2radians.

(c) Angle of-90oequivalent to-90×π

180= (i)-π2(ii)-π(iii)-3π2

(d) Angle of 60 oequivalent to 60×π

180= (one or more!)

(i)?60o (e) Angle of 75 oequivalent to 75×π

180= (one or more!)

(i)?75o

180o?π(ii)0.42π(iii)1.309radians.

(f) An angle of 106 oequivalent to 106×π

180= (one or more!)

(i)?106o

180o?π(ii)0.59π(iii)1.85radians.

(g) An angle of 166 oequivalent to 166×π

180= (one or more!)

(i)?166o

180o?π(ii)0.92π(iii)2.90radians.

(h) An angle of-466oequivalent to-466×π

180= (one or more!)

(i)?-466o

180o?π(ii)-2.59π(iii)-8.13radians.

Section 1. Definitions of the Trigonometric Functions (LECTURE NOTES 9)151

3.Converting Radians To Degrees.

1 radian =

180o
(a) Radianπequivalent toπ×180

π= (i)90o(ii)180o(iii)360odegrees.

(b) Radian 3π

2equivalent to3π2×180π= (i)90o(ii)270o(iii)360o

(c) Radian-3π

2equivalent to-3π2×180π=

(i)-90o(ii)-270o(iii)-360odegrees. (d) Radian 2πequivalent to 2π×180

π= (one or more)

(i)?2π

π?180o(ii)2(180o)(iii)360odegrees.

(e) Radian 5π

9equivalent to5π9×180π= (one or more)

(i)?5π/9

π?180o(ii)59(180o)(iii)100odegrees.

(f) Radian 7π

2equivalent to7π2×180π= (one or more)

(i)?7π/2

π?180o(ii)72(180o)(iii)630odegrees.

(g) Radian of 1.3 equivalent to 1.3×180

π= (one or more)

(i)?1.3

4.Evaluating trigonometric functions using the definitions.

sinθ=y rcscθ=ry,(y?= 0) cosθ=x rsecθ=rx,(x?= 0) tanθ=y x,(x?= 0) cotθ=xy,(y?= 0) xy (x,y) = (-6, 4) x yr II I IV

IIIθ

Figure 13.3 (Evaluating trigonometric functions at (x,y) = (-6,4)) (a) Sincex=-6,y= 4, r=⎷ x2+y2=?(-6)2+ 42≈(i)7.211(ii)8.211(iii)9.211

152Chapter 13. The Trigonometric Functions (LECTURE NOTES 9)

(b) sinθ=y sinθis (i)positive(ii)negative (c) cosθ=x cosθis (i)positive(ii)negative (d) tanθ=y tanθis (i)positive(ii)negative (e) cscθ=r cscθis (i)positive(ii)negative (f) secθ=r secθis (i)positive(ii)negative (g) cotθ=x y≈-64= (i)0.55(ii)-1.50(iii)-0.83 cotθis (i)positive(ii)negative (h)Where is(x,y) = (-6,4)? Point (x,y) = (-6,4) is in quadrant (i)I(ii)II(iii)III(iv)IV (i)What isθ?

θ= sin-1?y

r?≈sin-1?47.211?≈ (i)-43.31(ii)-53.31(iii)-33.31degrees

MODE DEGREE ENTER 2nd SIN

-14 / 7.211 ENTER orθ= tan-1?y x?≈tan-1?-46?≈ (i)-33.31(ii)-43.31(iii)-53.31degrees

2nd TAN

-1-4 / 6 ENTER Any of the other trigonometric functions such as cos -1, csc-1could also be used as well.

5.Evaluating an inverse trigonometric function.

45 - 45 - 90 triangleo oo

xy45 45
90
ooo r = 2 1_ x = 1 y = 1 0 y = 1 r = 2 1_ 45o45
o x = 1 90o

180 - 45 = 135ooo

Figure 13.4 (Evaluating inverse trigonometric functions

Find allθbetween 0 and 2πwhere sinθ=1

⎷2 Section 1. Definitions of the Trigonometric Functions (LECTURE NOTES 9)153 (a)Solve using special angles.Since sinθ=y r=1⎷2, soy= 1 andr=⎷2, this is a (i)30o-60o-90otriangle (ii)45o-45o-90otriangle also, since bothy= 1 and, of course, radiusr=⎷

2 are positive,

this meansθcould only be intwoquadrants: (i)I(ii)II(iii)III(iv)IV

Look at the figure.

Specificallyθ= (choose two!)

(i)45o(ii)135o180o-45o= 135o(iii)90o (b)Solve using calculator.since sinθ=y r=1⎷2; that is, since bothy= 1 andr=⎷2 are positive, this meansθcould only be intwoquadrants: (i)I(ii)II(iii)III(iv)IV

Look at the figure.

since sinθ=1 ⎷2,θ= sin-1?1⎷2?= (i)45o(ii)135o(iii)90o

MODE DEGREES ENTER 2nd SIN

-11 ⎷2 butθcould also be in quadrant II, so

θ= (i)45o(ii)135o180o-45o= 135o(iii)90o

6.Transforming graphs ofcostandsint.Constantsa,b,c,dtransform graphs of

bothasin(bx+c) +dandacos(bx+c) +din the following ways: •amplitudeaincreases (decreases) "height" of graph for|a|large (small) whena <0, graphs reflected inx-axis ("flipped"),

•constantb(assumeb >0) affects period;

graph ofy= sin(bx) looks likey= sinxbut with periodT=2π bif 0< b <1, period completed more rapidly (shorter period) thanb= 1 ifb >1, period completed more slowly (longer period) than whenb= 1 •horizontal shiftcmoves graphsleft(c >0) orright(c <0) •vertical shiftdmoves graphs up (d >0) or down (d <0)

154Chapter 13. The Trigonometric Functions (LECTURE NOTES 9)

ty 1 -1 (a) amplitude, a, and vertical shift, d--22ty 1 -1 (b) period, b, and horizontal shift, c y = cos(t) y = 2cos(t) amplitude doubled y = cos( /2) period doubled y = cos(t) y = cos(t - /2) horizontal shift right y = cos(t) - 2 vertical shift down

π π--22ππ πππ

Figure 13.5 (Transforming graphs of costand sint)

(GRAPH using Yquotesdbs_dbs14.pdfusesText_20

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