Note that this is Chapter 7 starting with the new edition
Section 6.3 Section 6.4 Section 6.5 Section 6.1-2 Section 6.7-8
Note that Test #1 includes Sections 6.3, 6.4 and 6.5 but not Sections 6.1 and 6.2.
This is due to a change in the order of these sections in the new edition of the text.
The logic of the original order has been preserved in tests for this course.
(These sections are 7.3 - 7.8 starting with 8th edition)
Section 6.3 (7.3 starting with 8th edition)
Establish each identity.
6. Sin `theta ( cot `theta + tan `theta ) = sec `theta
10. ( csc `theta - 1 ) ( csc `theta + 1 ) = cot^2(`theta)
12. ( csc `theta + cot `theta ) ( csc `theta - cot `theta ) = 1
18. csc ˆ4 `theta- csc^2(`theta) = cotˆ4 `theta + cot^2(`theta)
20. csc `theta - cot `theta =( sin `theta / ( 1 + cos `theta )
24. 1 - ( sin² `theta / ( 1- cos `theta ) ) = -cos `theta
30. ( cos `theta + 1/ ( cos `theta - 1) ) = ( 1+ sec `theta) /( 1- sec `theta )
36. ( 1 - cos `theta ) / ( 1 + cos `theta) = ( csc `theta - cot `theta )^2
40. sin `theta cos `theta / ( cos^2 (`theta) - sin^2( `theta) ) = tan `theta / ( 1- tan^2( `theta ))
42. (sin `theta — cos `theta +1) /( sin `theta + cos `theta – 1 ) = ( sin `theta +1) / cos `theta
48. sec`theta / (1 + sec `theta ) = ( 1 – cos `theta ) /sin^2( `theta )
50. (1- cot^2( `theta ) ) / ( 1+ cot^2(`theta) )+ 2 cos^2(`theta) = 0
54. tan `theta + cot `theta –sec `theta csc `theta = 0
60. (sec^2(`theta) – tan^2(`theta) + tan `theta) / sec `theta =sin `theta + cos `theta
66. ( cos `theta + sin `theta–sin³`theta) /sin`theta = cot `theta + cos`theta
70. ( 1 + cos `theta + sin `theta) / ( 1+ cos `theta–sin `theta )= sec `theta+ tan `theta
72. ( 2a sin `theta cos`theta)^2 + a^2(cos^2(theta) – sin^2 (`theta))^2=a^2
78. ln |tan`theta |=ln |sin`theta| –ln| cos`theta |
80. ln |sec`theta+tan`theta |+ln |sec`theta–tan`theta |=0
Section 6.4 (7.4 starting with 8th edition)
Find the exact value of each trigonometric function.
6. sin 105 `deg
10. tan(19 `pi /12)
12. cot(-5 `pi /12)
Find the exact value of each expression.
18. (tan 40 `deg –tan 10 `deg) / (1+tan 40 `deg tan 10 `deg)
20. cos (5 `pi /12) cos (7 `pi /12) – sin (5 `pi /12) sin (7 `pi /12)
Find the exact value of each of the following under the given conditions.
24. cos `alpha = `sqrt( 5 )/5, 0<`alpha < `pi /2; sin `beta = -4/5, - `pi /2<`beta<0
30. If cos `theta=¼,with `theta in quadrant IV, find the exact value of
Establish each identity.
36. Cos ( `pi + `theta ) = -cos `theta
40. cos (3 `pi / 2 + `theta ) = sin `theta
42. cos ( `alpha + `beta ) + cos (`alpha – `beta ) = 2 cos `alpha cos `beta
48. cos (`alpha + `beta)/cos (`alpha–`beta) = (1- tan `alpha tan `beta) / (1+ tan `alpha tan `beta)
50.cot(`alpha–`beta) = (cot`alpha cot`beta+1)/ (cot`beta–cot`beta)
54.cos(`alpha–`beta)cos(`alpha+`beta) = cos^2(`alpha) – sin^2(`beta)
60.If tan `alpha = x+1and tan`beta = x–1,show that 2 cot(`alpha–`beta) = x^2
Section 6.5 (7.5 starting with 8th edition)
For problems 6-12, use the information given about the angle `theta, with 0<=`theta<=2 `pi ,to find the exact value of
6.sin `theta=-`sqrt( 3 )/3, with 3`pi /2 < `theta < 2 `pi
10.sec`theta = 2, csc `theta < 0
12.cot`theta = 3, cos`theta < 0
use the half angle formulas to find the exact value of each of the trigonometric function.
18.sin 195 `deg
20. csc(7 `pi /8)
24.Develop a formula for cos 3`theta as a third- degree polynomial in the variable cos`theta.
Establish each identity.
30. (cot`theta–tan`theta) / (cot`theta+tan`theta) = cos (2`theta)
36 csc^2(`theta/2) = 2/ (1–cos`theta)
42 tan(`theta/2)= csc`theta–cot`theta
48.tan`theta+tan(`theta+120 `deg)+ tan(`theta+240 `deg)= 3 tan (3`theta)
50.If x = 2* tan`theta, express cos(2 `theta) as a function of x.
60.If tan `theta=a tan(`theta/3), express tan(`theta/3) in terms of a.
Section 6.1-2 (7.1-7.2 starting with 8th edition)
Find the exact value of each expression.
6. tan^-1 (-1)
10. sin^-1(-`sqrt( 3) /2 )
12. sin^-1 (-`sqrt( 2 )/2 )
Use a calculator to find the value of each expression rounded to two decimal placec.
18.sin^-1(1/8)
20. tan^-1 (-3)
24. sin^-1(`sqrt( 3) /5 )
Find the exact value of each expression.
30.cot[ sin^-1(-1/2)]
36. csc[cos?¹(-`sqrt( 3) / 2 )]
40. cos?¹[cos(-`pi /3)]
42. tan(cos^-1 ( 1/3) )
50 csc(tan^-1 (1/2) )
54. sin(sin^-1`sqrt( 3 )/2 +cos^-1 (1))
60. sec(tan^-1(4/3)+cot^-1(5/12))
66. cos(2 cos^-1(4/5))
72 .cos^2(1/2 * sin^-1(3/5))
Write each trigonometric expression as an algebraic expression containing u and v.
78.sin(sin^-1(u) - cos^-1(v))
80.cos(tan^-1(u) + tan^-1(v))
Establish each identity
84.Show that tan(sin^-1(v))=v/`sqrt( (1-v²) )
90. Show that tan^-1(v)+cot^-1(v)=`pi /2
Use a calculator to find the value of each expression rounded to two decimal places.
96.csc^-15
100. cot^-1(-1/2)
102. cot^-1(-8.1)
108. For what numbers x does sin(sin^-1(x))=x?
110. For what numbers x does sin^-1(sin x)=x?
114. Graph y=csc^-1(x).
Section 6.7-8 (7.7-7.8 starting with 8th edition)
Solve each equation on the interval 0<=2 <=2`pi
6. sin 2 =`sqrt( 2 )/2
10.tan( 2`theta) = -1
12.cot ( 2`theta/3 ) = -`sqrt( 3 )
Use a calculator to solve each equation on the interval 0<= `theta <= 2 `pi
18.cos `theta = 0.6
20.cot`theta = 2
24.csc`theta = -3
Solve each equation on the interval 0<=`theta<=2`pi
30. (cot `theta + 1)(csc`theta - 1/2) = 0
36.tan `theta = cot `theta
40.cos (2`theta) + cos )4`theta) = 0
42.sin (4`theta) -sin (6`theta) = 0
48.cos (2`theta) +5 cos(`theta) + 3 = 0
50.sec(`theta) = tan (`theta) + cot (`theta)
54. tan (2`theta) + 2 cos`theta=0
Use a graphing utility to solve each equation. Express the solution(s)correct to two decimal places.
66.19x+8 cosx=2
70.x^2+3sin x=0
72.x^2=x+3 cos 2x
84.The speed of yellow sodium light(wavelength of 589 nanometers) in a certain liquid is measured to be 1.9x10^8 meters per second. What is the index of refraction of this liquid, with respect to air, of sodium light?
Hint: the speed of light in air is approximately 2.99x10^8meters per second.