course Mth 164
24-23-836Assignment #2
6.2
3. For function y = f(x), for each x in the domain, there is exactly one element y in the range. T
9. Exact values can be found for the trigonometric functions of 60 deg. T
11. (sq. rt. of 3, ½) sin t = y = ½ , cos t = x = sq rt o f 3, tan t = y/x = sq. rt. of 3/3,
Csc t = 1/y = 2, sec t= 1/x = 2(sq. rt. of 3)/3, cot t = x/y = sq. rt. of 3
15. ( -sq. rt of 2/2, sq. rt. of 2/2) sin t = y = sq rt of 2/2, cos = t = x = -sq rt of 2/2
Tan t = y/x = -1 csc t= sq rt of 2 sec t = - sq rt of 2 cot t = -1
19. sin 11pi/2(180/pi) = 990 deg. Y= -1
21. tan 6 pi(180/pi) = 1080 deg. Tan = y/x = 0/0 = 0
27. sec(-pi)(180/pi) = -180 deg. = -1
33. sin(45 deg)cos(45 deg) = (sq. rt. of 2/2)*(sq. rt. of 2/2) = ½
39. 2 sin (pi/3) 3 tan (pi/6) = 2(sq. rt of 3 /2) 3(sq. rt of 3/3) = 0
45. tan pi cos 0 = 0-1 = -1
49. 2pi/3 = (-1/2, sq. rt. of 3/2) sin (2pi/3) = y = sq. rt. of 3/2 , cos (2pi/3) = x = -1/2,
tan (2pi/3) = y/x = -sq. rt. of 3, csc (2pi/3) = 1/y = 2(sq.rt. of 3)/3,
sec (2pi/3) = 1/x = -2 cot (2pi/3) = x/y = sq. rt. of 3/2
51. 210 deg. (pi/180) = (-sq. rt. 3/2, -1/2) sin (pi/180) = y = -1/2, cos (pi/180) = x = - sq. rt. of 3/2, tan (pi/180) = y/x = sq. rt. of 3/3, csc (pi/180) = 1/y = -2,
sec (pi/180) = -2(sq. rt. of 3)/3, cot = sq. rt. of 3
57. 405 deg. (pi/180) = 9 pi/4 = same as 45 deg.((sq. rt. of 2)/2, (sq. rt. of 2)/2)
Sin (pi/180) = y = (sq. rt. of 2)/2, cos (pi/180) = x = (sq. rt. of 2)/2,
tan (pi/180)= y/x = 1 Csc (pi/180) = 1/y = sq rt of 2,
sec (pi/180) = 1/x = sq. rt. of 2, cot t = x/y = 1
61.-45 deg.(pi/180) = (-pi/4) ((sq rt of 2)/2);( (-sq. rt of 2)/2)
Sin(-45 deg) = y = ((-sq. rt. of 2)/2) cos(-45 deg) = x =((sq. rt. of 2)/2)
Tan(-45 deg) = y/x = -1 csc(-45 deg) = 1/y = -(sq. rt. of 2)
sec(-45 deg) = 1/x = sq. rt. of 2 cot(-45 deg) = x/y = -1
63. 5pi/2(180/pi) = 450 deg. (0, 1) sin(5pi/2) = y = 1 cos((5pi/2) = x = 0
Tan(5pi/2) = y/x = undef. Csc(5pi/2) = 1/y = 1 sec(5pi/2) = 1/x= undef.
Cot.(5pi/2) = x/y = 0
67. sin28 deg. = .27
73. sin pi/10 = .31
79. sin 1 = .84
Find the exact values of the 6 trig. Functions given a pt. on the terminal side of an angle:
83. (-3,4) (-3)^2 +4^2 = r^2
9+16 = r^2
25 = r^2
5 = r
Sin theta = y/r = 4/5 csc theta = r/y = 5/4
Cos theta = x/r = -3/5 sec theta = r/x = 5/-3
Tan theta = y/x = 4/-3 cot theta = x/y = -3/4
85. (2, -3) x^2+y^2 = r^2
2^2+(-3)^2 = 13
R = sq. rt. of 13
Sin theta = y/r = -3/(sq. rt. of 13) csc theta = r/y = (sq. rt. of 13)/-3
Cos theta = x/r = 2/(sq. rt. of 13) sec theta = r/x = (sq. rt. of 13)/2
Tan theta = y/x = -3/2 cot theta = x/y = 2/-3
91. (1/3, -1/4) (1/3)^2+(-1/4)^2 = r^2
1/9 + 1/16 = r^2
sin theta = y/r = 5/12 csc theta = r/y = -5/3
cos theta = x/r = 4/5 sec theta = r/x = 5/4
tan theta = y/x = -3/4 cot theta = x/y = -4/3
97. *****
103. f(theta) = sin (theta). Find the exact value of each function if (theta) = 60 deg.
60 deg. = pi/180 = pi/3 = sin(60) = (sq. rt. of 3)/2
109. f(2 theta) = sin( 2 theta) = sin(120) = 120(pi/180) = 2pi/3 = (sq. rt. of 3)/2
115. theta f(theta) = sin(theta)/theta
.5 .9589
.4 .9735
.2 .9933
.1 .9983
.01 .999983
.001 .9999983
.0001 .99999983
.00001 .999999983
As the value of theta approaches 0, the value of of sin(theta)/theta approaches 1.
121. a = length of base g = 32 ft/sec/ acceleration, inclined plane w/ base a = 10 ft.
a. theta = 30 deg. t = sq. rt. of : (2*10)/(32 sin(30) cos(30)) = 1.20 sec.
b. theta = 45 deg. t = sq. rt. of : (2*10)/(32sin(45)cos(45)) = 1.12 sec.
c. theta = 60 deg. t = sq. rt. of: (2*10)/(32sin(60)cos(45)) = 1.20 sec.
127. t = 1 = app. 60 deg.
Sin theta = (sq. rt. of 3)/2 = app. .8660 csc theta = 2(sq. rt. of 3)/3 = app. 1.1547
Cos theta = .5 sec theta = 2
Tan theta = sq. rt. of 3 = app. 1.73 cot theta = (sq. rt. of 3)/3 = app. .5774
Sec. 6.3
3. f(x) = sq. rt. of x is even T or F ?
False because it is not symmetric w/ respect to the y-axis when graphed
9. All the trigonometric functions are periodic, with period 2pi false because tangent and cotangent functions are periodic w/ period pi
. find the exact value:
11. sin 405 deg. 405 360 = 45 same as 45 deg. 17pi/4 = (pi/4 + 4pi) = sin (pi/4) = (sq. rt. of 2)/2 = .7071
15. csc 450 deg. 450-360= 90 deg. csc (90) = 1
21. tan(21pi) = tan (20pi + pi) = tan pi =.0549
27. sin theta greater than 0, cos theta less than 0 - quadrant II
33. sec theta less than 0, sin theta greater than 0 - quadrant II
Given sin and cos, find the value of the remaining 4 trig. Functions:
35. sin theta =(-3/5) csc theta = -5/3
Cos theta= 4/5 sec theta = 5/4
Tan theta = -3/4 cot theta = -4/3
39. sin theta = ½ csc theta = 2
Cos theta = (sq. rt. of 3)/2 sec theta = 2*(sq. rt. of 3)/3
Tan theta = (sq. rt. of 3) cot theta = sq. rt. of 3
Find the exact value of the remaining trig. Functions:
43. sin theta =12/13, theta in quadrant II:
Csc theta greater than 0, others neg.
Sin theta = 12/13 csc theta = 13/12
Sin^2(theta) + cos^2(theta) =1
(12/13)^2+cos^2(theta) = 1
144/169 + cos^2(theta) = 1
Cos(theta) = -5/13 = x (because cos(theta) is greater than 0 in quadrant 2)
Sec(theta) = -13/5
Tan(theta) = -12/5 cot(theta) = -5/12
45. cos(theta) = -4/5 , theta in quadrant III
tan^2(theta) + 1 = sec^2(theta)
tan^2(theta) +1 = (-5/4)^2 = 25/16
tan(theta) = Ύ
sin(theta) = -3/5 cos(theta) = -5/3
cos(theta) = -4/5 sec(theta) = -5/4
tan(theta) = Ύ cot(theta) = 4/3
51.sin(theta) = 2/3, tan less than 0 , quadrant II
Cot^2(theta) +1 = csc^2(theta) csc(theta) = 3/2
Cot^2(theta) +1 = (3/2)^2
Cot(theta) = +/-sq. rt. of 5/4
Cos(theta) = -.7454 sec(theta) = 1/.7454
Tan(theta) = -.8944 cot(theta) = -sq. rt. of-5/4
57. tan(theta) = -1/3, sin (theta) is greater than zero , quadrant II
Sin(theta) = 1 csc(theta) = 1
Cos(theta) = -3 sec(theta) = -1/3
Tan(theta) = -1/3 cot(theta) = 3
Tan(theta) = sin(theta)/cos(theta) = -1/3 = y/x = sin = y = 1
Cos = x= -3
Use even-odd properties to find exact values.
59. sin(-60 deg) odd function 60(pi/180) = pi/3 = (-sq. rt. of 3)/2
63. sec(-60 deg) even = pi/3 = 1/x = 2
69. cos(-pi/4) even = (sq. rt. of 2)/2
75. sec(-pi/6) = pos. 1/x = 2(sq. rt. of 3)/3
Use prop. of trig. Functions to find the exact value
81. tan40 deg. (sin40 deg./cos40 deg.)
-sin(theta)/cos(theta) = -tan(theta) = tan40 deg. tan 40 deg. = 0
79. sin80 deg. csc80 deg. sin = y csc = 1/y sin/cos = y/y = 1
87. sin(-20 deg)/cos 380 deg. + tan 200 deg. = -sin20 deg. /cos380 deg. + tan(200) = 0
**not sure how to do this one w/out calculator
93. find the exact value of sin 1 deg. + sin 2 deg. sin 3 deg.
sin 358 deg. +sin359 deg. = 0
.0174524064 -.0174524064
.0348994967 -.0348994967
.0523359562 -.0523359562
97. For what numbers theta is f(theta) = tan(theta) not defined ? All real numbers, except odd integer multiples of pi/2 .
99. For what numbers theta is f(theta) = sec(theta) not defined ? All real numbers, except odd integer multiples of pi/2.
105. What is the range of the secant function ? All real numbers greater than or equal to 1 or less than or equal to -1.
111. Secant function odd, even, neither ? even, graph symmetric
117. If f(theta) = sec(theta) and f(a) = -4, find the exact value of
A) f(-a) = f(-(-4)) = 4 b) f(a) + f(a+2pi) +(a+4pi) = -8.8584
****( I checked the answer in the back of the book and I know it is -12. I read in the book that you can ignore 2pi and multiples of 2pi because they represent complete revolutions, but Im not sure I understand why.
Think of the unit circle. If you start at a point and go completely around the circle you end up where you started.
A complete trip around the circle corresponds to a change of 2 pi radians in angular position.
n complete trips around the circle would correspond to a change of n * 2 pi = 2 pi n radians in angular position.
If f(theta) is a function whose value is determined by the 'theta' position, i.e., angular position, on the unit circle, then if theta changes by 2 pi radians, or by 2 pi n radians (where n is an integer), your position on the circle won't change. The function f(theta) won't know the difference. So we can safely say that
f(theta) = f(theta + 2 pi n).
So if f(a) = 4, it follows that f(a+2pi) = 4 and f(a+4pi) = 4, so that
f(a) + f(a+2pi) +(a+4pi) = 4 + 4 + 4 = 12.
123. Show that the period of f(theta) =sin(theta) is 2 pi
**** dont really understand this If f(0) = sin(0) +p then p= 0
If f(pi/2) = sin(pi/2) + p then p= -.0274
Not sure if this is right and if so I dont understand what it means
sin(theta) is the y coordinate of the unit-circle point for position theta. So as we've seen in the qa exercises that preceded this text assignment,
sin(pi/6) = 1/2
sin(pi/4) = sqrt(2)/2
sin(pi/3) = sqrt(3)/2
sin(pi/2) = 1
sin(2 pi / 3) = sqrt(3) / 2
etc..
Adding 2 pi to theta doesn't change where you are on the unit circle. So it doesn't change the value of sin(theta).
Thus sin(theta) has period 2 pi.
129.Prove the reciprocal identities given if formula 2:
1) csc(theta) = 1/sin(theta) csc(theta) = 1/y, sin(theta) = y
2)sec(theta) = 1/cos(theta) sec(theta) = 1/x, cos(theta) = x
3) cot(theta) = 1/tan(theta) , tan(theta) = y/x, cot(theta) = 1/y/x = x/y
135. Explain how to find the value cos(-45 deg) using even-odd. Prop.:
Cos is an even prop. cos(-45 deg.) = cos(45 deg) 45(pi/180) = pi/4 =
(sq. rt. of 2)/2
5.2 online problems
6. (1, -1)
Sin = y = -1 csc = 1/y = -1
Cos = x= 1 sec = 1/x = 1
Tan = y/x = -1 cot = x/y = -1
10.(-.3, -.4)
Sin = -.4 csc = -2.5
Cos = -.3 sec = -10/3
Tan = 4/3 cot = Ύ
Without using a calculator, find:
12. sin30 deg. cos 45 deg. = ½ - (sq. rt. of 2)/2 = app. 1.09
18. sec(30 deg) * cot(45 deg.) = (2*sq. rt. of 3/3)/3
20. 5cos(90 deg.) 8 sin(270 deg.) = 5*0 8*(-1) = 8
24. tan(pi/3) + cos(pi/3) = (sq. rt. of 3) + ½ = 2.23
31. sec(pi) csc(pi/2) = -2
Find the 6 trig. functions:
36. pi/3 (180/pi) = -60 deg. (1/2, (-sq. rt. of 3/2))
Sin(-pi/30) = (-sq. rt. of 3)/2 csc(-pi/3) = -2(sq. rt. of 3)/3
Cos(-pi/3) = ½ sec(-pi/3) = 2
Tan(-pi/3) = (-sq. rt. of 3) cot(-pi/3) = (-sq. rt. of 3)/3
40. 3pi (180/pi) = 540 deg. same as 180 deg. (-1, 0)
Sin(3pi) = 0 csc(3pi) = undef.
Cos.(3pi) = -1 sec(3pi) = -1
Tan(3pi) = 0 cot(3pi) = undef.
42. -270 deg. (pi/180) = -3pi/2 (0, -1)
Sin(-270) = -1 csc(-270) = -1
Cos(-270) = 0 sec(-270) = undef.
Tan(-270) = undef. Cot(-270) = 0
If theta = 60 deg. find the exact value of
72. cos(theta) = 60 (pi/180) = pi/3 = ½
78. cos(2 theta) = cos(120 deg) = 120(pi/180) = -1/2
80. 2cos(theta) = 2(cos theta) = 2(.5) = 1
84. Find the exact value of tan 60 deg. and tan 150 deg.
Tan(60) (pi/180) = pi/3 = sq. rt. of 3
Tan(150) (pi/180) = 5pi/6 = (-sq. rt. of 30/3
90. If cos (theta) = 2/3 find sec(theta) x = 2/3 sec(theta) = 1/x = 3/2
Sec. 5.3 online problems
6. sec(540 ) deg. = 540(pi/180) = 3 pi sec(theta + 2 pi) (ignore 2 pi) = sec(pi) = -1
10. sin(9pi/4) = (8pi/4 + pi/4) = 2pi + pi/4 (ignore 2pi) = sin(pi/4) = (sq. rt. of 2)/2
12. csc(9pi/2) = (8pi/2 + pi/2) = 4 pi + pi/2)(ignore 4 pi) = csc(pi/2) = 1
18 sin theta less than 0, cos theta greater than 0 : IV
20. cos theta greater than 0 , tan theta greater than 0 : I
24. csc theta greater than 0, cos theta less than 0 : II
In the next problem sin theta and cos theta are given . Find the exact value of each of the 4 remaining trig. functions.
30. sin(theta) = 2(sq. rt. of 2)/3 csc(theta)= 3( sq. rt. of 2)/4
Cos(theta) = -1/3 sec (theta) = -3
Tan(theta) = -2(sq. rt. of 2) cot(theta) = (-sq. rt. of 2)/4
36. sin(theta) = -5/13 , theta in quadrant 3 y = -5, r= 13 x = -12
Csc(theta) = -13/5
Cos(theta) = -12/13 sec(theta) = 13/-12
Tan(theta) = 13/12 cot(theta) = 12/13
40. sin(theta) = -2/3 , pi less than theta is less than 3pi/2 Quadrant III
Csc(theta) = -3/2
Cos(theta) = (-sq. rt. of 5)/3 sec(theta) = -3(sq. rt. of 5)/5
Tan(theta) = 2(sq. rt. of 5)/5 cot(theta) = (sq. rt. of 5)/2
42. cos(theta) -1/4, tan(theta) greater than 0 , Quadrant III
x= -1, y = sq. rt. of 15, r= 4
sin(theta) = (-sq. rt. of 15)/4 csc(theta) = -4(sq. rt. of 15)/15
sec(theta) = -4
tan(theta) = - sq. rt. of 15 cot(theta) = (sq. rt. of 15)/15
48. sec(theta) = -2, tan(theta) greater than 0 Quadrant III
R= 2 x = -1 y = sq. rt. of 3
Sin(theta) = (-sq. rt. of 3)/2 csc(theta) =- 2(sq. rt. of 3)/3
Cos(theta) = -1/2
Tan(theta) = sq. rt. of 3 cot(theta) = (sq. rt. of 3)/3
Use even-odd properties to find exact value of each expression.
50.cos (-30 deg) = even cos(-theta) = cos(theta) =cos(30) = 30(pi/180) = pi/6 = (sq. rt. of 3)/2
54. csc(-30 deg) = odd csc(-theta) = -(-csc theta) = 30(pi/180) = (sq. rt. of 3)/2
60. sin(-pi/3) = odd = sin(-theta) = (-sin theta) = (-pi/3)(180/pi) = 60 deg. =
(sq. rt. of 3)/2
66. csc(-pi/3) = odd = csc(-theta) = -csc(theta) = csc(pi/3)(180/pi) = 60 deg. =
(sq. rt. of 3)/2
Find the exact value of each expression.
70. tan(-6pi)+cos(9pi/4) = tan(-pi)+ cos(8pi/4+pi/4) = undef. + sq. rt. of 2/2 = undef.
72. cos(-17pi/4)-sin(-3pi/2) = (-sq. rt. of 2)/2 + 1
78. cot 20 deg. cos 20 deg/sin20deg. = cot = x/y, cos = x, sin = y x/y x/y=0
80. If cos(theta) =.2, find the value of cos(theta) +(theta + 2pi) + cos(theta + 4pi) = ignore 2pi, 4pi .2 + .2+.2 = .6
84. Domain of cosine function: all real numbers
90. Range of the cosine function: all real numbers from -1 to 1 inclusive
96. cosine function even, symmetric w/ respect to x-axis
100 cosecant function odd , symmetric w/ respect to x-axis
102. If f(x) = cos(x) and f(a) = Ό, find the exact value of
F(-a) cos(-1/4)
F(a) +f(a+2pi) + f(a-2pi) (ignore 2pi) (1/4)+(1/4)+(1/4) =.75
110. Use the periodic and even-odd properties to show that the range of the cotangent function is the set of all real numbers.
Range = y values
Cot = x/y odd function
Cot(theta + pi) = cot (theta)
Your work in general looks good; however use the qa's and queries to submit you assignments before submitting problems in this format. Many of the questions you asked here are answered in the Query for this assignment.
The format you are using here is very good for submitting questions about specific problems that aren't covered in the q_a_ or query.
You've asked a couple of questions in the above and I've inserted explanations, which I hope will be helpful. Let me know if you have additional questions.